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Paley–Wiener properties for spaces of entire functions

Paley–Wiener properties for spaces of entire functions We deduce Paley–Wiener results in the Bargmann setting. At the same time we deduce characterisations of Pilipovic´ spaces of low orders. In particular we improve the characterisation of the Gröchenig test function space H = S , deduced in Toft (J Pseudo-Differ Oper Appl 8:83–139, 2017). Mathematics Subject Classification Primary 46F05 · 32A25 · 32A36; Secondary 35Q40 · 30Gxx 1 Introduction Paley–Wiener theorems characterize functions and distributions with certain restricted supports in terms of estimates of their Fourier–Laplace transforms. For example, let d d f be a distribution on R and let B (0) ⊆ R be the ball with center at origin and radius r . Then f is supported in B (0) if and only if 0 r N r | Im(ζ )| d | f (ζ )|  ζ  e ,ζ ∈ C , for some N ≥ 0. Furthermore, f is supported in B (0) and smooth, if and only if − N r | Im(ζ )| d | f (ζ )|  ζ  e ,ζ ∈ C , for every N ≥ 0. Joachim Toft joachim.toft@lnu.se Elmira Nabizadeh elmira.nabizadehmorsalfard@lnu.se Christine Pfeuffer christine.pfeuffer@mathematik.uni-halle.de Department of Mathematics, Linnæus University, Växjö, Sweden Department of Mathematics, Martin-Luther-University of Halle-Wittenberg, Halle (Saale), Germany 0123456789().: V,-vol 20 Page 2 of 34 E. Nabizadeh et al. A similar approach for ultra-regular functions of Gevrey types and corresponding ultra-distribution spaces can be done. In fact, let s > 1, D (R ) be the set of all Gevrey distributions of order s and let E (R ) be the set of all smooth functions with Gevrey regularity s. (See [6] and Sect. 2 for notations.) Then it can be proved that f ∈ D (R ) is supported in B (0), if and only if r | Im(ζ )|+r |ζ | d | f (ζ )|  e ,ζ ∈ C , for every r > 0. Furthermore f ∈ E (R ) is supported in B (0), if and only if s r r | Im(ζ )|−r |ζ | d | f (ζ )|  e ,ζ ∈ C , for some r > 0. We observe that s in the latter result can not be pushed to be smaller, because if s ≤ 1, it does not make any sense to discuss compact support properties of D (R ) and E (R ). In the paper we consider analogous Paley–Wiener properties after the Fourier– Laplace transform above is replaced by the reproducing kernel  for the Bargmann transform, and the image spaces are replaced by suitable subspaces of entire functions on C . These subspaces were considered in [4,12] and are given by d d d d A (C ) = A (C ), A (C ) = A (C ), r , 0, r , σ σ σ σ r >0 r >0 (1.1) d d d d A (C ) = A (C ), A (C ) = A (C ), s r ,s 0,s r ,s r >0 r >0 1 d d when σ> 0 and 0 < s < , where A (C ) and A (C ) are the sets of all entire r , r ,s F on C such that 2σ 1 σ +1 1−2s r |z| r (logz) | F (z)|  e respective | F (z)|  e . The spaces in (1.1) appear naturally when considering the Bargmann transform images of extended classes of Fourier invariant Gelfand–Shilov spaces, called Pilipovic´ spaces (see [4,12]). If (z,w) is the scalar product of z,w ∈ C , then the reproducing kernel of the Bargmann transform is given by −d (z, · )−| · | ( F )(z) = π  F , e when F is a suitable function or (ultra-)distribution. If R|z|−|z| 1 d z → F (z)e ∈ L (C ), R >0(1.2) Paley–Wiener properties for spaces of entire functions Page 3 of 34 20 holds and dλ(w) is the Lebesgue measure on C , then −d (z,w)−|w| ( F )(z) = π F (w)e dλ(w). A recent Paley–Wiener result with respect to the transform  and image spaces ∞ d  d ∞ d (1.1)isgiven in [12], where it is proved that if L (C ) = E (C ) ∩ L (C ), then d ∞ d d (E (C )) =  (L (C )) = A (C ). (1.3) A A c 1 ∞ d  d Evidently, L (C ) ⊆ E (C ), and the gap between these spaces are rather large. It might therefore be somewhat surprising that the first equality holds in (1.3). In Sect. 3 we improve (1.3) in different ways. Firstly we show that we may replace ∞ d ∞ d L (C ) in (1.3) with the smaller space L (C ) given by A,c ∞ d ∞ L (C ) = L (K ), A,c A,c where L (K ) is the set of all F · χ , where F is analytic in a neighbourhood of the A,c compact set K and χ is the characteristic function of K . Secondly, we may replace d  d E (C ) with the set E (C ) of all compactly supported Gevrey distributions of order s > 1. Summing up we improve (1.3)into d ∞ d d (E (C )) =  (L (C )) = A (C ), s > 1. (1.3) A A s A,c 1 In Sect. 3 we also deduce various kinds of related mapping properties when A (C ) ∞ d in (1.3) is replaced by any of the spaces in (1.1). More precisely, let χ ∈ L (C ) be non-negative, radial symmetric in each complex variable z and bounded from below by a positive constant near origin. Then we prove d d 1 (A (C ) · χ) = A (C ), σ ∈ ( , 1), σ = , A  0 0, σ σ 2 2σ − 1 d d 1 (A (C ) · χ) = A (C ), σ ∈ (0, ), σ = , A   0 σ σ 1 − 2σ d d (A (C ) · χ) = A (C ), s ∈[0, ), A s s and similarly for σ = and when A and A are replaced by A and A , s 0, 0,s σ σ respectively. (Cf. Theorems 3.2–3.8 and Propositions 3.9–3.11.) Finally, in Sect. 4 we use the results in Sect. 3 to deduce characterizations of Pilipovic´ spaces of small orders. We remark that some continuations of the present analysis, involving larger spaces of analytic functions or formal power series expan- sions, can be found in [13]. 20 Page 4 of 34 E. Nabizadeh et al. 2 Preliminaries In this section we recall some basic facts. We start by discussing Pilipovic´ spaces and some of their properties. Then we recall some facts on modulation spaces. Finally we recall the Bargmann transform and some of its mapping properties, and introduce suitable classes of entire functions on C . 2.1 The Pilipovic´ spaces In order to define Pilipovic´ spaces we recall the definition of Hermite functions. We recall that the Hermite function of order α ∈ N is defined by |x | d 1 2 − |α| |α| − α −|x | 4 2 2 h (x ) = π (−1) (2 α!) e (∂ e ). It follows that d 1 |x | − |α| − − 4 2 2 h (x ) = π (2 α!) e p (x ), α α for some polynomial p on R , which is called the Hermite polynomial of order α.The Hermite functions are eigenfunctions to the Fourier transform, and to the Harmonic oscillator H ≡|x | − which acts on functions and (ultra-)distributions defined on R . More precisely, we have H h = (2|α|+ d)h , H ≡|x | − . d α α d It is well-known that the set of Hermite functions is a basis for S (R ) and an 2 d 2 d orthonormal basis for L (R ) (cf. [10]). In particular, if f ∈ L (R ), then 2 2 f  = |c ( f ,α)| , 2 d h L (R ) α∈N where f (x ) = c ( f ,α)h (2.1) h α α∈N is the Hermite series expansion of f , and c ( f ,α) = ( f , h ) 2 d (2.2) h α L (R ) is the Hermite coefficient of f of order α ∈ R . In order to define the full scale of Pilipovic´ spaces, their order s should belong to the extended set R = R ∪{  ; σ ∈ R }, + σ + Paley–Wiener properties for spaces of entire functions Page 5 of 34 20 of R , with extended inequality relations as s < < s and  < 1 σ 2 σ σ 1 2 when s < ≤ s and σ <σ .(Cf.[12].) 1 2 1 2 For such s and r ∈ R we set 1 1 1 1 2s 2s −( ·α +···+ ·α ) ⎪ 1 r 1 r d ⎨ 1 d e , s ∈ R \{ }, ϑ (α) ≡ − (2.3) r ,s α 2σ r α! , s =  , ⎪ σ α 1 d r , s = ,α ∈ N and 1 1 1 2s 1 2s ( ·α +···+ ·α ) ⎪ 1 r 1 r d ⎨ 1 d e , s ∈ R \{ }, ϑ (α) ≡ (2.4) r ,s 2σ r α! , s =  , ⎪ σ α d r , s = ,α ∈ N . Definition 2.1 Let s ∈ R = R ∪{0}, and let ϑ and ϑ be as in (2.3) and (2.4). r ,s r ,s d  d (1) H (R ) consists of all finite Hermite series expansions (2.1), and H (R ) consists of all formal Hermite series expansions in (2.1); d d 2 d (2) if s ∈ R , then H (R ) (H (R )) consists of all f ∈ L (R ) such that s 0,s |c ( f , h )|  ϑ (α) h α r ,s d d holds true for some r ∈ R (for every r ∈ R ); + + d  d (3) if s ∈ R , then H (R ) (H (R )) consists of all formal Hermite series expan- 0,s sions in (2.1) such that |c ( f , h )|  ϑ (α) h α r ,s d d holds true for every r ∈ R (for some r ∈ R ). + + d d The spaces H (R ) and H (R ) are called Pilipovic ´ spaces of Roumieu respectively s 0,s d  d Beurling type of order s, and H (R ) and H (R ) are called Pilipovic ´ distribution s 0,s spaces of Roumieu respectively Beurling type of order s. d d Remark 2.2 Let S (R ) and (R ) be the Fourier invariant Gelfand–Shilov spaces of s s order s ∈ R and of Rourmeu and Beurling types respectively (see [12] for notations). Then it is proved in [8,9] that d d H (R ) = (R ) ={0}, s > , 0,s s d d H (R ) = (R ) ={0}, s ≤ , 0,s s 2 20 Page 6 of 34 E. Nabizadeh et al. d d H (R ) = S (R ) ={0}, s ≥ s s and d d H (R ) = S (R ) ={0}, s < . s s In Proposition 2.3 below we give further characterisations of Pilipovic´ spaces. Next we recall the topologies for Pilipovic´ spaces. Let s ∈ R , r > 0, and let f  and  f   be given by s;r H s;r f  ≡ sup |c ( f ,α)ϑ (α)|, s ∈ R , (2.5) H h r ,s s;r α∈N and f   ≡ sup |c ( f ,α)ϑ (α)|, s ∈ R . (2.6) H h r ,s s;r α∈N when f is a formal expansion in (2.1). Then H (R ) consists of all expansions s;r (2.1) such that  f  is finite, and H (R ) consists of all expansions (2.1) such s;r s;r d  d that  f   is finite. It follows that both H (R ) and H (R ) are Banach spaces s;r s;r s;r under the norms f → f  and f → f   , respectively. H H s;r s;r d d We let the topologies of H (R ) and H (R ) be the inductive respectively pro- s 0,s jective limit topology of H (R ) with respect to r > 0. In the same way, the s;r d  d topologies of H (R ) and H (R ) are the projective respectively inductive limit 0,s topology of H (R ) with respect to r > 0. It follows that all the spaces in Definition s;r d  d 2.1 are complete, and that H (R ) and H (R ) are Fréchet spaces with semi-norms 0,s f → f  and f → f  , respectively. s;r s;r The following characterisations for Pilipovic´ spaces can be found in [12]. The proof is therefore omitted. d d Proposition 2.3 Let s ∈ R ∪{0} and let f ∈ H (R ). Then f ∈ H (R )( f ∈ + 0,s d ∞ d H (R )), if and only if f ∈ C (R ) and satisfies H f  ∞ sup < ∞ N 2s h N ! N ∈N for every h > 0 (for some h > 0). From now on we let d |x | − − 4 2 φ(x ) = π e . (2.7) Paley–Wiener properties for spaces of entire functions Page 7 of 34 20 2.2 Spaces of entire functions and the Bargmann transform Let  ⊆ C be open. Then A() is the set of all analytic functions in . If instead ⊆ C is closed, then A() is the set of all functions which are analytic in an open neighbourhood of . In particular, if z ∈ C is fixed, then A({z }) is the set of all 0 0 complex-valued functions which are defined and analytic near z . We shall now consider the Bargmann transform which is defined by the formula d 1 1 − 2 4 2 (V f )(z) = π exp − (z, z+| y| ) + 2 z, y f ( y) dy, d 2 2 d 2 d when f ∈ L (R ) (cf. [1]). We note that for all f ∈ L (R ), then the Bargmann transform V f of f is the entire function on C , given by (V f )(z) = A (z, y) f ( y) dy, d d or (V f )(z) = f , A (z, · ), (2.8) d d where the Bargmann kernel A is given by d 1 1 − 2 4 2 A (z, y) = π exp − (z, z+| y| ) + 2 z, y . Here d d z,w= z w , when z = (z ,..., z ) ∈ C and w = (w ,...,w ) ∈ C , j j 1 d 1 d j =1 and otherwise · , · denotes the duality between test function spaces and their corresponding duals. We note that the right-hand side in (2.8) makes sense when d d f ∈ S (R ) and defines an element in A(C ), since y → A (z, y) can be inter- 1/2 d d preted as an element in S (R ) with values in A(C ). 1/2 2 d It was proved in [1] that f → V f is a bijective and isometric map from L (R ) 2 d 2 d d 2 d to the Hilbert space A (C ) ≡ B (C ) ∩ A(C ), where B (C ) consists of all mea- surable functions F on C such that F  2 ≡ | F (z)| dμ(z) < ∞. (2.9) −d −|z| d Here dμ(z) = π e dλ(z), where dλ(z) is the Lebesgue measure on C .We 2 d 2 d recall that A (C ) and B (C ) are Hilbert spaces, where the scalar products are given by 2 d ( F , G) ≡ F (z)G(z) dμ(z), F , G ∈ B (C ). (2.10) C 20 Page 8 of 34 E. Nabizadeh et al. 2 d If F , G ∈ A (C ), then we set  F  = F  and ( F , G) = ( F , G) . 2 2 2 2 A B A B Furthermore, Bargmann showed that there is a convenient reproducing formula on 2 d A (C ). More precisely, let (z,w) ( F )(z) ≡ F (w)e dμ(w), (2.11) R|z|−|z| 1 d when z → F (z)e belongs to L (C ) for every R ≥ 0. Here d d (z,w) = z w , when z = (z ,..., z ) ∈ C and w = (w ,...,w ) ∈ C , j j 1 d 1 d j =1 d d is the scalar product of z ∈ C and w ∈ C . Then it is proved in [1,2] that  is 2 d 2 d the orthogonal projection of B (C ) onto A (C ). In particular,  F = F when 2 d F ∈ A (C ). In [1] it is also proved that V h = e , where e (z) ≡ √ , z ∈ C . (2.12) d α α α α! 2 d In particular, the Bargmann transform maps the orthonormal basis {h } d in L (R ) α∈N 2 d bijectively into the orthonormal basis {e } d of monomials in A (C ). Hence, there α∈N is a natural way to identify formal Hermite series expansion by formal power series expansions F (z) = c( F,α)e (z), (2.13) α∈N by letting the series (2.1) be mapped into c ( f ,α)e (z). (2.14) h α α∈N 2 d 2 d It follows that if f , g ∈ L (R ) and F , G ∈ A (C ), then ( f , g) 2 d = c ( f ,α)c (g,α), h h L (R ) α∈N ( F , G) = c( F,α)c(G,α). (2.15) 2 d A (C ) α∈N Here and in what follows, ( · , · ) and ( · , · ) denote the scalar products in 2 d 2 d L (R ) A (C ) 2 d 2 d L (R ) and A (C ), respectively. Furthermore, c ( f ,α) = c( F,α) when F = V f . (2.16) h d We now recall the following spaces of power series expansions given in [12]. Paley–Wiener properties for spaces of entire functions Page 9 of 34 20 Definition 2.4 Let s ∈ R = R ∪{0}, and let ϑ and ϑ be as in (2.3) and (2.4). r ,s r ,s d d  d (1) A (C ) consists of all analytic polynomials on C , and A (C ) consists of all formal power series expansions on C in (2.13); d d 2 d (2) if s ∈ R , then A (C ) (A (C )) consists of all F ∈ L (C ) such that s 0,s |c( F , h )|  ϑ (α) α r ,s holds true for some r > 0 (for every r > 0); d  d (3) if s ∈ R , then A (C ) (A (C )) consists of all formal power series expansions s 0,s in (2.13) such that |c( F , h )|  ϑ (α) r ,s holds true for every r > 0(forsome r > 0). Let f ∈ H (R ) with formal Hermite series expansion (2.1). Then the Bargmann transform V f of f is defined to be the formal power series expansion (2.14). It 2 d follows that V agrees with the earlier definition when acting on L (R ), that V d d d  d is linear and bijective from H (R ) to A (C ), and restricts to bijections from the 0 0 spaces d d  d  d H (R ), H (R ), H (R ) and H (R ) (2.17) 0,s s s 0,s to d d  d  d A (C ), A (C ), A (C ) and A (C ) (2.18) 0,s s s 0,s respectively, when s ∈ R . We also let the topologies of the spaces in (2.18)be inherited from the spaces in (2.17). d  d d  d If s ∈ R , f ∈ H (R ), g ∈ H (R ), F ∈ A (C ) and G ∈ A (C ), then s s s s ( f , g) and ( F , G) are defined by the formula (2.15). It follows that (2.16) 2 d 2 d L (R ) A (C ) d d holds for such choices of f and F. Furthermore, the duals of H (R ) and A (C ) s s d  d can be identified with H (R ) and A (C ), respectively, through the forms in (2.15). s s The same holds true with H , H , A , and A 0,s 0,s 0,s 0,s in place of H , H , A , and A , s s s s respectively, at each occurrence. In order to identify the spaces of power series expansions above with spaces of analytic functions, we let 1 1 1−2s 1−2s ⎪ r (logz ) + ··· + r (logz ) , s < , 1 1 d d 2σ 2σ σ +1 σ +1 M (z) = 1,r ,s r |z | + ··· + r |z | , s =  ,σ > 0, 1 1 d d σ ⎪ 2 1 1 ⎩ |z| 1 s s − (r |z | + ··· + r |z | ), s ≥ , 1 1 d d 2 2 20 Page 10 of 34 E. Nabizadeh et al. M (z), s = , 1,r ,s 0 2 M (z) = 1,r ,s 2 2 r |z | + ··· + r |z | , s = , 1 1 d d 2σ 2σ σ −1 σ −1 r |z | + ··· + r |z | , s =  ,σ > 1, 1 1 d d σ M (z) = 2,r ,s 2 1 1 |z| 1 s s + (r |z | + ··· + r |z | ), s ≥ , 1 1 d d 2 2 M (z), s = , 2,r ,s M (z) = (2.19) 2,r ,s 1 2 2 r |z | + ··· + r |z | , s = , 1 1 d d when r ∈ R and z ∈ C . For conveniency we set M = M . r 1, ,r + 1 By [12] we have the following. The proof is therefore omitted. 0 0 Proposition 2.5 Let M ,M ,M and M be as in (2.19) when s ∈ R and 1,r ,s 2,r ,s 1,r ,s 2,r ,s r ∈ R . Then − M d d ∞ d d 1,r ,s A (C ) ={ F ∈ A(C ) ; Fe ∈ L (C ) for every r ∈ R }, s > 0, 0,s d d − M ∞ d d 1,r ,s A (C ) ={ F ∈ A(C ) ; Fe ∈ L (C ) for some r ∈ R }, s ≥ 0, d d − M ∞ d d 2,r ,s A (C ) ={ F ∈ A(C ) ; Fe ∈ L (C ) for every r ∈ R }, s > , s + d d − M ∞ d d 2,r ,s A (C ) ={ F ∈ A(C ) ; Fe ∈ L (C ) for some r ∈ R }, s > , 0,s + d d  d A (C ) = A(C ) and A (C ) = A({0}). 0, 1 1 Next we recall the link between the Bargmann transform and the short-time Fourier transform f → V f with window function φ given by (2.7), defined by −i· ,ξ V f (x,ξ) ≡ f , φ(·− x )e . Let S be the dilation operator given by 1 1 − − 2 2 (SF )(x,ξ) = F (2 x , −2 ξ), (2.20) 1 2d when F ∈ L (R ). Then it follows by straight-forward computations that loc d 1 2 2 1 1 (|x | +|ξ | ) −i x ,ξ 2 2 2 2 (V f )(z) = (V f )(x + iξ) = (2π) e e V f (2 x , −2 ξ) d d φ d 1 2 2 (|x | +|ξ | ) −i x ,ξ  −1 2 2 = (2π) e e (S (V f ))(x,ξ), (2.21) or equivalently, d 1 1 1 2 2 − − (|x | +|ξ | ) −i x ,ξ /2 − − 2 4 2 2 V f (x,ξ) = (2π) e e (V f )(2 x , −2 ξ) φ d d |·| − −i x ,ξ /2 − 2 2 = (2π) e S(e (V f ))(x,ξ) (2.22) We observe that (2.21) and (2.22) can be formulated as −1 V = U ◦ V , and U ◦ V = V , d V φ d φ V Paley–Wiener properties for spaces of entire functions Page 11 of 34 20 2d  d where U is the linear, continuous and bijective operator on D (R )  D (C ), given by d 1 1 1 2 2 (|x | +|ξ | ) −i x ,ξ 2 2 2 2 (U F )(x,ξ) = (2π) e e F (2 x , −2 ξ). (2.23) Let D (z ) be the polydisc d,r 0 { z = (z ,..., z ) ∈ C ;|z − z | < r , j = 1,..., d }, 1 d j 0, j j with center and radii given by d d z = (z ,..., z ) ∈ C and r = (r ,..., r ) ∈ (0, ∞) . 0 0,1 0,d 1 d Then A(C ) = A( D (z)), A({0}) = A( D (z )). d,r d,r 0 d d r ∈(0,∞) r ∈(0,∞) 2.3 Hilbert spaces of power series expansions and analytic functions The spaces in Definition 2.4 can also be described by related unions and intersections of Hilbert spaces of analytic functions and power series expansions as follows. (See also [12].) d d Let ϑ be a weight on N , ω be a weight on C , and let ⎛ ⎞ 1 ⎝ ⎠ F  ≡ |c( F , α)ϑ (α)| (2.24) 2 d A (C ) [ϑ ] α∈N when F ∈ A (C ) is given by (2.13), and F  ≡ | F (z)ω(2 z)| dμ(z) (2.25) 2 d A (C ) (ω) d 2 d  d when F ∈ A(C ).Welet A (C ) be the set of all F ∈ A (C ) such that  F  is [ϑ ] 0 A [ϑ ] 2 d d finite, and A (C ) be the set of all F ∈ A(C ) such that  F  2 is finite. It follows (ω) A (ω) that these spaces are Hilbert spaces under these norms. If ϑ and ω are related to each others as 2 α ϑ(α) = ω (r ) r dr (2.26) α! and |z| 2 2 ω(z) = e ω (|z | ,..., |z | ), (2.27) 0 1 d 20 Page 12 of 34 E. Nabizadeh et al. for some suitable weight ω on R , then the following multi-dimensional version 2 d 2 d of [7, Theorem (4.1)] shows that A (C ) = A (C ) with equal norms. Here we [ϑ ] (ω) identify entire functions with their power series expansions at origin. Consequently, 2 d 2 d the Bargmann transform is bijective and isometric from H (R ) to A (C ) for such [ϑ ] (ω) choices of ϑ and ω. We omit the proof since the result is an immediate consequence of [12, Theorem 3.5]. d d Theorem 2.6 Let α ∈ N and ω be a positive measurable function on R .Alsolet d d ϑ and ω be weights on N and C , respectively, related to each others by (2.26) and (2.27), and such that |α| ϑ(α), α ∈ N , (2.28) (α!) 2 d 2 d holds for every r > 0. Then A (C ) = A (C ) with equality in norms. [ϑ ] (ω) In our situation, the involved weights should satisfy a split condition. In one dimen- sion, (2.26), (2.27) and (2.28) take the forms 2 α ϑ (α ) = ω (r ) r dr ,α ∈ N, (2.26) j j 0, j j α ! j R |z | ω (z ) = e ω (|z | ), z ∈ C (2.27) j j 0, j j j and |α | ϑ (α ), r > 0,α ∈ N. (2.28) j j j (α !) Lemma 2.7 Let ω be weights on R , ω be weights on C and ϑ be weights on 0, j + j j N such that (2.26) –(2.28) hold, j = 1,..., d , and set ω (z) ≡ ω (z ), 0 0, j j j =1 z ≡ (z ,..., z ) ∈ C .If ϑ and ω are given by (2.26), and (2.27), then 1 d ϑ(α) = ϑ (α ), α = (α ,...,α ) ∈ N (2.29) j j 1 d j =1 and ω(z) = ω (z ), z = (z ,..., z ) ∈ C . (2.30) j j 1 d j =1 Paley–Wiener properties for spaces of entire functions Page 13 of 34 20 α 1 Proof By [12, Theorem 3.5] and its proof, it follows that ω · r ∈ L (R ) for all 0, j + j ∈{1,..., d} and α ∈ N. Hence, Fubini’s theorem gives ⎛ ⎞ 1 2 d 1 1 2 α 2 ⎝ ⎠ ϑ(α) = ω (r ) r dr = ω (r ) r dr 0 0, j j j α! α! R 0 j =1 = ϑ (α ), j j j =1 and (2.29) follows. The assertion (2.30) follows from the definitions. 2.4 A test function space introduced by Gröchenig In this section we recall some comparison results deduced in [12], between a test function space, S (R ), introduced by Gröchenig in [5] to handle modulation spaces with elements in spaces of ultra-distributions. The definition of S (R ) is given as follows. d |x | − − d d 4 2 Definition 2.8 Let φ(x ) = π e . Then S (R ) and S (R ) consist of all f ∈ C G d ∗ ∞ d  d  d S (R ) such that f = V F,for some F ∈ L (R ) ∩ E (R ) and F ∈ E (R ), respectively. It follows that f ∈ S (R ), if and only if d 1 2 − − |x − y| i x ,η 2 2 f (x ) = (2π) F ( y,η)e e dydη, (2.31) 2d ∞ d  d for some F ∈ L (R ) ∩ E (R ). Remark 2.9 By the identity (V h, F ) = (h, V F ) and the fact that the map ( f ,φ) → d d 2d ∗ V f is continuous from S (R ) × S (R ) to S (R ), it follows that f = V F is d  2d uniquely defined as an element in S (R ) when F ∈ S (R ) (cf. [3]). In particular, d d the space S (R ) in Definition 2.8 is well-defined, and it is evident that S (R ) ⊆ G C S (R ). The following is a restatement of [12, Lemma 4.9]. The result is essential when deducing the characterizations of Pilipovic´ spaces in Sect. 4. ∞ d  d ∗ Lemma 2.10 Let F ∈ L (C ) ∪ E (C ). Then the Bargmann transform of f = V F is given by  F , where A 0 √ √ d 1 2 2 3 (|x | +|ξ | ) −i x ,ξ 4 2 F (x + iξ) = (2π ) F ( 2x , − 2ξ)e e . (2.32) d d Moreover, the images of S (R ) and S (R ) under the Bargmann transform are C G given by ∞ d  d  d {  F ; F ∈ L (C ) ∩ E (C ) } and {  F ; F ∈ E (C ) }, (2.33) A A respectively. 20 Page 14 of 34 E. Nabizadeh et al. The next results follow from [12, Theorem 4.10]. The proof is therefore omitted. d d d Proposition 2.11 It holds S (R ) = S (R ) = H (R ). C G Due to the image properties for the spaces in Proposition 2.11 under the Bargmann transform, the next result is equivalent with the previous one. Proposition 2.12 The sets in (2.33) are equal to A (C ). In the next section we extend Propositions 2.11 and 2.12 by proving that the conclu- sions in Proposition 2.12 hold for suitable smaller and larger sets than those in (2.33). We also deduce similar identifications for other Pilipovic´ spaces and their Bargmann images. 3 Paley–Wiener properties for Bargmann–Pilipovic´ spaces In this section we consider spaces of compactly supported functions with interiors in d  d A (C ) or in A (C ). We show that the images of such functions under the reproducing kernel  are equal to A (C ), for some other choice of s ≤  . In the first part we state A s 1 the main results given in Theorems 3.2–3.4. They are straight-forward consequences of Propositions 3.9, where more detailed information concerning involved constants are given. Thereafter we deduce results which are needed for their proofs. Depending of the choice of s, there are several different situations for characterizing A (C ). This gives rise to a quite large flora of main results, where each one takes care of one situation. 2 2 First we note that if A = A , then a split of the variables in the weight ω in [ϑ ] (ω) 2 2 A induce a split of the variables in ϑ in A . (See Lemma 2.7.) (ω) [ϑ ] In order to present the main results, it is suitable to make the following definition. ∞ d Definition 3.1 Let t , t ∈ R be such that t ≤ t . Then the function χ ∈ L (C ) is 1 2 1 2 called positive, bounded and radial symmetric with respect to t and t , if the following 1 2 conditions are fulfilled: ∞ d • χ ∈ L (C ) ∩ E ( D (0)) is non-negative; • χ(z ,..., z ) = χ (|z |,..., |z |) for some function χ ; 1 d 0 1 d 0 • χ ≥ c on D (0) for some constant c > 0. The set of positive, bounded and radial symmetric functions with respect to t and t 1 2 ∞ d ∞ d is denoted by R (C ), and R (C ) is defined by t ,t 1 2 ∞ d ∞ d R (C ) ≡ R (C ). t ,t 1 2 t ≤t ∈R 1 2 3.1 Main results We begin with characterizing the largest spaces in our investigations, which appears when s =  , and then proceed with spaces of decreasing order. First we recall that 1 Paley–Wiener properties for spaces of entire functions Page 15 of 34 20 d d elements in A (C ) and A (C ) fulfill conditions of the forms s 0,s 2σ 2σ σ +1 σ +1 r |z | +···+r |z | 1 1 d d | F (z)|  e (3.1) when s =  and 1 1 1−2s 1−2s r (logz ) +···+r (logz ) 1 1 d d | F (z)|  e (3.2) when s ∈[0, ). Theorem 3.2 Let F ∈ A(C ), σ = 1 and s > 1. Then the following conditions are equivalent: (1) F ∈ A (C ); (2) (3.1) holds for some r ∈ R ; d d ∞ d (3) For some r ∈ R and every r ∈ R with r ≤ r and every χ ∈ R (C ), there 0 0 + + r ,r exists F ∈ A( D (0)) such that F =  ( F · χ); 0 r A 0 d d ∞ d (4) For some r ∈ R and every r ∈ R with r < r and some χ ∈ R (C ), there 0 0 + + r ,r exists F ∈ A( D (0)) such that F =  ( F · χ); 0 r A 0 d ∞ d (5) There exists F ∈ E (C ) ∩ L (C ) such that F =  F ; 0 A 0 (6) There exists F ∈ E (C ) such that F =  F . 0 A 0 Remark 3.3 Since d ∞ d  d  d E (C ) ∩ L (C ) ⊆ E (C ) ⊆ E (C ), Theorem 3.2 still holds true after E has been replaced by E in (6). d ∞ d Theorem 3.4 Let F ∈ A(C ), σ = 1 and χ ∈ R (C ). Then the following is true: (i) The following conditions are equivalent: (1) F ∈ A (C ); 0, (2) (3.1) holds for every r ∈ R ; (3) There exists F ∈ A(C ) such that F =  ( F · χ); 0 A 0 d d (ii) The map F →  ( F · χ) from A(C ) to A (C ) is a homeomorphism. A 0, The next result deals with the case when s =  with σ ∈ ( , 1). d ∞ d Theorem 3.5 Let F ∈ A(C ), χ ∈ R (C ), σ ∈ ( , 1) and let σ = . 2σ − 1 Then the following is true: (i) The following conditions are equivalent: 20 Page 16 of 34 E. Nabizadeh et al. d d (1) F ∈ A (C ) (F ∈ A (C )); 0, σ σ (2) (3.1) holds for some (for every) r ∈ R ; d d (3) There exists F ∈ A (C ) (F ∈ A (C )) such that F =  ( F · χ); 0 0 A 0 0, σ σ 0 0 d d  d (ii) The mappings F →  ( F · χ) from A (C ) to A (C ) and from A (C ) 0, σ σ σ 0 0 to A (C ) are homeomorphisms. 0, The next result deals with the case when s =  with σ = . d ∞ d Theorem 3.6 Let F ∈ A(C ), σ = and χ ∈ R (C ). Then the following is true: (i) The following conditions are equivalent: d d (1) F ∈ A (C ) (F ∈ A (C )); 0, σ σ (2) (3.1) holds for some (for every) r ∈ R ; d d (3) There exists F ∈ A (C ) (F ∈ A (C )) such that F =  ( F · χ); 0 0 0,1/2 A 0 0,1/2 d d (ii) The mappings F →  ( F · χ) from A (C ) to A (C ) and from 0,1/2 1/2 d d A (C ) to A (C ) are homeomorphisms. 0,1/2 0, 1/2 The next result deals with the case when s =  with σ ∈ (0, ). d ∞ d 1 Theorem 3.7 Let F ∈ A(C ), χ ∈ R (C ), σ ∈ (0, ) and let σ = . 1 − 2σ Then the following is true: (i) The following conditions are equivalent: d d (1) F ∈ A (C ) (F ∈ A (C )); 0, σ σ (2) (3.1) holds for some (for every) r ∈ R ; d d (3) There exists F ∈ A (C ) (F ∈ A (C )) such that F =  ( F · χ); 0  0 0, A 0 σ σ 0 0 d d d (ii) The mappings F →  ( F · χ) from A (C ) to A (C ) and from A (C ) A   0, σ σ σ 0 0 to A (C ) are homeomorphisms. 0, In the next result we consider the case when s ∈[0, ) is real. d ∞ d 1 Theorem 3.8 Let F ∈ A(C ), χ ∈ R (C ),s ∈[0, ). Then the following is true: (i) The following conditions are equivalent: d d (1) F ∈ A (C ) (F ∈ A (C )); s 0,s (2) (3.2) holds for some (for every) r ∈ R ; d d (3) There exists F ∈ A (C ) (F ∈ A (C )) such that F =  ( F · χ); 0 s 0 0,s A 0 d d d (ii) The mappings F →  ( F · χ) from A (C ) to A (C ) and from A (C ) to A s s 0,s A (C ) are homeomorphisms. 0,s Paley–Wiener properties for spaces of entire functions Page 17 of 34 20 The previous theorems are essentially consequences of Propositions 3.9–3.11, where more detailed information about involved constants are given. d d Proposition 3.9 Let F ∈ A(C ), σ = 1,s > 1 and r ∈ R . Then the following conditions are equivalent: (1) For some r ∈ R such that r < r, (3.1) holds with r in place of r ; 0 0 0 d d (2) For some t ∈ R such that t < r , every t ∈ R with t ≤ t < r and every 1 1 2 1 2 + + ∞ d χ ∈ R (C ), there exists F ∈ A( D (0)) such that F =  ( F · χ); 0 r A 0 t ,t 1 2 (3) For some F ∈ A( D (0)) it holds F =  ( F · χ ); 0 r A 0 D (0) ∞ d (4) For some F ∈ E ( D (0)) ∩ L (C ) it holds F =  F ; 0 r A 0 (5) For some r ∈ R such that r < r , there exists F ∈ E ( D (0)) such that 0 0 0 r + s 0 F =  F . A 0 Theorems 3.5–3.7 essentially follow from the following proposition. 1 d Proposition 3.10 Let τ> ,r , t , t ∈ R be such that t ≤ t , and let χ ∈ 1 2 1 2 ∞ d R (C ). Then the following is true: t ,t 1 2 (1) Let F ∈ A(C ) be such that 2 2 2τ +1 2τ +1 r |z | +···+r |z | 0,1 1 0,d d | F (z)|  e (3.3) d d holds for some r ∈ R such that r < r . Then for some r ∈ R such that 0 0 0 + + r < r , there exists F ∈ A(C ) such that F =  ( F · χ) and 0 0 A 0 2 2 2τ −1 2τ −1 R |z | +···+ R |z | 0,1 1 0,d d | F (z)|  e , (3.4) where 2τ +1 2τ −1 2τ − 1 2r − 2τ −1 R = t ; (3.5) 2 2τ + 1 (2) Let r , R ∈ R be such that r < r and 0 0 0 2τ +1 2τ −1 2τ − 1 2r − 2τ −1 R = t , (3.6) 2 2τ + 1 d d F ∈ A(C ) be such that (3.4) hold and let F =  ( F · χ). Then F ∈ A(C ) 0 A 0 and satisfies (3.3) for some r ∈ R such that r < r. 0 0 Theorem 3.8 follows from the following two propositions, where the first one concerns the case when s > 0 and the second one make a more detailed explanation of the case s = 0, i. e. the case of analytic polynomials. 1 d ∞ d Proposition 3.11 Let s ∈ (0, ),r ∈ R , and let χ ∈ R (C ). Then the following is true: 20 Page 18 of 34 E. Nabizadeh et al. (1) Suppose F ∈ A(C ) satisfies 1 1 1−2s 1−2s r (logz ) +···+r (logz ) 0,1 1 0,d d | F (z)|  e (3.7) for some r ∈ R such that r < r . Then there is an F ∈ A(C ) such that 0 0 0 F =  ( F · χ) and A 0 1 1 1−2s 1−2s r (logz ) +···+r (logz ) 0,1 1 0,d d | F (z)|  e (3.8) for some r ∈ R such that r < r; 0 0 (2) Suppose F ∈ A(C ) satisfies (3.8) for some r ∈ R such that r < r , and let 0 0 0 F =  ( F · χ). Then F ∈ A(C ) and satisfies (3.7) for some r ∈ R such A 0 0 that r < r. ∞ d Proposition 3.12 Let χ ∈ R (C ) and let N ≥ 0 be an integer. Then the following is true: (1) Suppose F ∈ A(C ) is given by F (z) = c( F,α)z , (3.9) |α|≤ N where {c( F,α)} ⊆ C. Then there is an F ∈ A(C ) such that F =  ( F · |α|≤ N 0 A 0 χ) and F (z) = c( F ,α)z , (3.10) 0 0 |α|≤ N where {c( F ,α)} ⊆ C and satisfies c( F ,α) = 0 when c( F,α) = 0; 0 |α|≤ N 0 (2) Suppose F ∈ A(C ) satisfies (3.10) for some {c( F ,α)} ⊆ C, and let F = 0 0 |α|≤ N ( F · χ). Then F ∈ A(C ) and satisfies (3.9) for some {c( F,α)} ⊆ C A 0 |α|≤ N such that c( F,α) = 0 when c( F ,α) = 0. 3.2 Preparing results and their proofs For the proofs of Propositions 3.9–3.11 and thereby of Theorems 3.2–3.8 we need some preparatory results. Because the proof of Proposition 3.9 needs some room, we put parts of the statement in the following separate proposition. At the same time we slightly refine some parts concerning the image of compactly supported elements in L under the map  . Proposition 3.13 Let s > 1,r , r ∈ R be such that r < r and suppose that either 0 0 F ∈ E ( D (0)), F ∈ E ( D (0)), F ∈ E ( D (0)) or F ∈ L ( D (0)). 0 r 0 r 0 r 0 r s 0 0,s 0 0 Then F =  F ∈ A(C ) and satisfies A 0 r |z |+···+r |z | 1 1 d d | F (z)|  e . Paley–Wiener properties for spaces of entire functions Page 19 of 34 20 Proof By the inclusions E → E → E → E 0,s+ε s 0,s when ε> 0, it suffices to consider the case when F ∈ E ( D (0)) or F ∈ L ( D (0)) 0 r 0 r s 0 hold. Let r = r. First suppose that F ∈ E ( D (0)) holds, choose r ∈ R such that 2 0 r 1 s 0 + 2 2 −(| y| +|η| ) (z, y+i η) d r < r < r , ( y,η) = e and let  ( y,η) = e . By identifying C 0 1 2 z 2d with R and using the fact that F ∈ E ( D (0)) we obtain 0 r s 0 D ( ) z L ( D (0)) −d 1 | F (z)|= π | F , |  sup (3.11) A 0 0 z |α| 2d h α! α∈N 2d 2d for every h > 0. We also have  ∈ E (R )→ E (R ), which implies 1 1/2 0,s |α| α s D  ∞  h α! L ( D (0)) for every h > 0. Furthermore, α (z, y+i η) r |z |+···+r |z | 1,1 1 1,d d | D  ( y,η)|=|m (z)e |≤|m (z)|e , y + i η ∈ D (0), z α α r where α +α j d+ j d 2d m (z) = z , z ∈ C ,α ∈ N . j =1 By choosing h = 4 and h = 1 above, and letting y + i η ∈ D (0), Leibnitz rule 1 2 r gives −|α| −s α 4 α! | D ( )( y,η)| −|α| −s γ α−γ ≤ 4 α! | D  ( y,η)|| D ( y,η)| γ ≤α ⎛ ⎞ −|α| −s s r |z |+···+r |z | 1,1 1 1,d d ⎝ ⎠ 4 α! |m (z)|(α − γ)! e γ ≤α ⎛ ⎞ −|α| −s r |z |+···+r |z | 1,1 1 1,d d ⎝ ⎠ 4 |m (z)|γ ! e γ ≤α ⎛ ⎞ γ γ j d+ j |z | |z | j j r |z |+···+r |z | 1,1 1 1,d d ⎝ ⎠ ≤ sup · e . (3.12) s s γ ! γ ! γ ≤α j d+ j j =1 20 Page 20 of 34 E. Nabizadeh et al. In the last inequality we have used that the number of terms in the sums are bounded |α| k by 2 , and that ≤ 2 when n, k are non-negative integers such that k ≤ n. By combining (3.12) with the estimate γ γ j j 1 |z | (|z | ) j j s s|z | = ≤ e , γ ! γ ! j j we get ∞ 1 1 D ( ) z L ( D (0)) r s s 1 2s(|z | +···+|z | ) r |z |+···+r |z | 1 d 1,1 1 1,d d sup  e e |α| 2d h α! α∈N r |z |+···+r |z | 2,1 1 2,d d e . In the last inequality we have used the fact that r < r and s > 1. From the latter 1 2 estimate and (3.11) we obtain r |z |+···+r |z | 2,1 1 2,d d | F (z)|  e , A 0 and the result follows when F ∈ E ( D (0)). 0 r s 0 ∞ d Suppose instead that F ∈ L ( Q) holds, where Q = D (0) ⊆ C , and let 0 r Q = D (0) ⊆ C. Then j r 2, j (z,w) | F (z)|  | F (z)||e | dλ(w) A 0 0 |z ||w | r |z |+···+r |z | j j 2,1 1 2,d d ≤ F  ∞ e dλ(w )  e , 0 L j j =1 and the result follows in this case as well. In the next lemma we give options on compactly supported functions which are mapped on the basic monomials, e by the operator  . α A d ∞ d Lemma 3.14 Let t , t ∈ R be such that t ≤ t , χ ∈ R (C ), and let χ be such 1 2 1 2 0 + t ,t 1 2 that χ (|z |,..., |z |) = χ(z ,..., z ).If 0 1 d 1 d F (z) = ς z χ(z), α,χ α with −1 1 2 −d −|u| 2α d ς = 2 α! χ (u)e u u ··· u du , ={ u ∈ R ; u ≤ t }, α 0 1 d t (3.13) then the following is true: Paley–Wiener properties for spaces of entire functions Page 21 of 34 20 (1)  F = e ; A α,χ α (2) for some constant C > 0 which only depends on χ  ∞ , c in Definition 3.1 and the dimension d , it holds ⎛ ⎞ ⎛ ⎞ d d 1 2 1 −1 −2 −2α |t | −2 −2α ⎝ ⎠ ⎝ ⎠ 2 2 C t (α + 1) t α! ≤ ς ≤ Ce t (α + 1) t α! . j α j 2, j 2 1, j 1 j =1 j =1 (3.14) Proof By using polar coordinates in each complex variable when integrating we get −d α (z,w)−|w| ( F )(z) = π ς w χ(w)e dλ(w) A α,χ α −d α −|u| = π ς I (u, z)u χ (u)e u ··· u du, (3.15) α α 0 1 d where ⎛ ⎞ d d −i θ i α,θ  z u e j j ⎝ ⎠ I (u, z) = e e dθ = I (u , z ) (3.16) α α j j [0,2π) j =1 j =1 with 2π −i θ i α θ z u e j j j j I (u , z ) = e e dθ . α j j j By Taylor expansions we get ∞ k k −ikθ 2π j z u e j j i α θ j j I (u , z ) = e dθ α j j j k! k=0 α α j j ∞ k k 2π z u 2π z u j j j j i (α −k)θ j j = e dθ = k! α ! k=0 By inserting this into (3.15) and (3.16) we get α α u z 2 −d d α −|u| ( F )(z) = π ς (2π) u χ (u)e u ··· u du A α,χ α 0 1 d α! 1 2 d − 2α −|u| = 2 ς α! u χ (u)e u ··· u du e (z) = e (z) α 0 1 d α α and (1) follows. 20 Page 22 of 34 E. Nabizadeh et al. Since χ is non-negative and fullfills χ ≥ c on D (0) we get −1 2 1 |t | 2α ς  e α! u u ··· u du α 1 2 ⎛ ⎞ ⎛ ⎞ −1 2α +2 d d 2 1 2 1 1, j |t | |t | −2 −2α 2 2 ⎝ ⎠ ⎝ ⎠ 2 2 = e α!  e α! (t (α + 1)) t 1, j 1 2α + 2 j =1 j =1 which gives the right inequality in (3.14). By the support properties of χ we also have −1 2α ς  α! t u ··· u du , α 1 2 and the left inequality in (3.14) follows by similar arguments, and (2) follows. The next lemma shows that we may estimate entire functions by different Lebesgue norms. We omit the proof, since the result follows from [11, Theorem 3.2]. Lemma 3.15 Suppose s,τ ∈ R and r , r ∈ R are such that 1 1 s < ,τ > − and r < r . 2 2 Let p, q ∈[1, ∞],F ∈ A(C ) and set 2 2 2τ +1 2τ +1 M (z) = r |z | + ··· + r |z | 1,r 1 1 d d and 1 1 1−2s 1−2s M (z) = r (logz ) + ··· + r (logz ) . 2,r 1 1 d d Then − M − M j ,r j ,r F · e  p d   F · e  q d . L (C ) L (C ) The next lemma relates Lebesgue estimates of entire functions with estimates on corresponding Taylor coefficients. Here we let the Gamma function on R be defined by (x ,..., x ) = (x ), d 1 d j j =1 where  is the Gamma function on R . + Paley–Wiener properties for spaces of entire functions Page 23 of 34 20 1 d Lemma 3.16 Let τ> − ,r ∈ R ,M be the same as in Lemma 3.15, ω(z) = 1,r 1 2 |z| − M (z) d d 1,r e ,z ∈ C , and let α = (1,..., 1) ∈ N .Alsolet ((2τ + 1)(α + α )) −(2τ +1)(α+α ) ϑ (α) = (2τ + 1)(2r ) . α! If F ∈ A(C ) is given by (2.13), then ⎛ ⎞ 1 − M 2 1,r ⎝ ⎠ F · e  2 d = π |c( F,α)ϑ (α)| , L (C ) α∈N 2 d 2 d and A (C ) = A (C ) with equality in norms. [ϑ ] (ω) Proof Since 2τ +1 − M (z) −r |z | 1,r j j e = e , j =1 Lemma 2.7 shows that we may assume that d = 1, giving that r = r and α = 1. 1 0 2 d 2 d In view of Theorem 2.6 we have A (C ) = A (C ) with equality in norms, [ϑ ] (ω ) when 2τ +1 −2rt α ϑ(α) = e t dt . α! 2τ +1 By u = 2rt as new variables of integration we obtain −(2τ +1)(α+1) −u α(2τ +1)+2τ ϑ(α) = (2τ + 1)(2r ) e u du α! ((2τ + 1)(α + 1)) −(2τ +1)(α+1) = (2τ + 1)(2r ) , α! which implies that θ = θ , and the result follows. 1,r We also need the following version of Stirling’s formula. Lemma 3.17 Let α ≥ 0 be an integer and let τ> − . Then ((2τ + 1)(α + 1)) (2τ +1)·α τ 2τ (2τ + 1) (α + 1) · α! . (3.17) α! 20 Page 24 of 34 E. Nabizadeh et al. Lemma 3.17 follows by repeated applications of Stirling’s formula and the standard limit lim 1 + = e t →∞ for every x ∈ R. In order to be self-contained we present the arguments. τ τ Proof The result is obviously true for α = 0. For α ≥ 1wehave α  (α + 1) .A combination of the the latter relations and Stirling’s formula gives (2τ +1)·α+2τ + α ((2τ + 1)(α + 1)) ((2τ + 1)α + 2τ ) e α+2τ(α+1) α+ α! e 2 (2τ +1)·α+2τ + 1 2τ (2τ +1)·α+2τ + 2τ(α+1) (2τ + 1) 1 + α (2τ + 1)α 2τ(α+1) 2τ(α+1) (2τ +1)·α (2τ +1)·α τ 2τ (2τ + 1)  (2τ + 1) (α + 1) α! , (3.18) 2τα and the result follows. Proposition 3.9 essentially follows from the following lemma. Lemma 3.18 Let τ> − and r , r ∈ R be such that r < r . Then the following is 0 0 true: (1) If 2τ +1 ·α 2r −τ |c( F,α)|  α! , (3.19) 2τ + 1 then 2 2 2τ +1 2τ +1 r |z | +···+r |z | 1 1 d d | F (z)|  e ; (3.20) (2) If 2 2 2τ +1 2τ +1 r |z | +···+r |z | 0,1 1 0,d d | F (z)|  e (3.21) then 2τ +1 ·α 2r −τ |c( F,α)|  α! (3.22) 2τ + 1 Proof Let ϑ be the same as in Lemma 3.16 and r ∈ R be such that r < r < r. r 1 0 1 First we prove (1). Suppose that (3.19) holds and let α = (1,..., 1) ∈ N .Alsolet M be the same as in Lemma 3.15. Then Lemmas 3.15 and 3.16 give 1,r ⎛ ⎞ − M − M 1,r 1,r ⎝ ⎠ F · e  ∞ d   F · e  2 d  c( F,α)ϑ (α) L (C ) L (C ) 1 α∈N Paley–Wiener properties for spaces of entire functions Page 25 of 34 20 ⎛ ⎞ 2τ +1 2 (α+α ) ⎝ ⎠ = c( F,α)ϑ (α) α∈N sup c( F,α)ϑ (α) (α + 1) r j α∈N j =1 2τ +1 ·α 2τ + 1 sup |c( F,α)| α! . 2r d 0 α∈N Here the second inequality follows from the fact that ⎛ ⎛ ⎞ ⎞ (2τ +1)·(α+α ) τ r ⎝ ⎝ ⎠ ⎠ (α + 1) j =1 α∈N is convergent since r < r , and the fifth relation follows from Lemma 3.17.This 0 1 implies that (3.20) holds and (1) follows. Next assume that (3.21) holds. By Lemma 3.15 we get − M − M 1,r 1,r 0 1 F · e  ∞ d   F · e  2 d L (C ) L (C ) ⎛ ⎞ 1 ⎝   ⎠ c( F,α)ϑ (α) ≥ sup |c( F,α)| ϑ (α) r r 1 1 α∈N α∈N ⎛ ⎞ 2τ +1 2τ + 1 τ ⎝ ⎠ sup |c( F,α)| (α + 1) α! 2r α∈N j =1 2τ +1 2τ + 1 sup |c( F,α)| α! , 2r α∈N where the third inequality follows from Lemma 3.17. This gives (2). Proposition 3.11 mainly follows from the following result. d 1 d Lemma 3.19 Let r , r ∈ R be such that r < r, s ∈ (0, ) and let F ∈ A(C ). Then 0 0 the following is true: (1) if (3.7) holds, then 1−2s 1 1 2s 1 − 2s 2s 2s −( R |α | +···+ R |α | ) 1 1 d d |c( F,α)|  e , R = s ; (3.23) r 20 Page 26 of 34 E. Nabizadeh et al. (2) if R ∈ R is given by 1−2s 1 1 2s 1 − 2s 2s 2s −( R |α | +···+ R |α | ) 0,1 1 0,d d |c( F,α)|  e , R = s , (3.24) then (3.7) holds with r in place of r . Proof Let r , r , r ∈ R be such that r < r < r < r . By Lemma 2.7 we may 1 2 3 1 2 3 assume that d = 1, and by Lemma 3.15 the result follows if we prove 1 1 1−2s R ·α 2s 2 −r (logz) 2 sup |c( F,α)e |  | F (z)e | dλ(z) α∈N C 2s R ·α 2 sup |c( F,α)e | , (3.25) α∈N where R , R , R ∈ R satisfy 1 2 3 + 1−2s 2s 1 − 2s R = , j = 1, 2, 3. By Theorem 2.6 we have 1−2s −r (logz) 2 2 | F (z)e | dλ(z)  |c( F,α)ϑ (α)| , α∈N where ∞ 1 1−2s −r (logt ) α ϑ (α) = e t dt . 2α! Let −r (log t ) α θ = > 1 and g (t ) = e t . r ,α 1 − 2s In order to prove (3.25), we need to show that 1 1 2s 2s R α R α 2 1 e  ϑ (α)  e , (3.26) which shall be reached by modifying the proof of (15) in [4]. We have 2 (r −r )(log t ) ϑ (α)  e g (t ) dt r r ,α e Paley–Wiener properties for spaces of entire functions Page 27 of 34 20 (r −r )(log t ) sup(g (t )) e dt  sup(g (t )). r ,α r ,α 1 1 t ≥e e t ≥e By straight-forward computations it follows that g (t ) attains its global maximum r ,α for 1−2s 2s t = exp , (3.27) r ,α θ r and that 1 1 1− 2s − 2 R ·α 2s 1 2s 2s g (t ) = exp r θ (θ − 1)α = e , r ,α r ,α 1 1 and the second inequality in (3.26) follows. In order to prove the first inequality in (3.26), we claim that for some c which is independent of α we have c θ (r −r )(log t ) 2 r ,α 1 −  e . (3.28) r ,α In fact, by (3.27) it follows that −1 lim α t =∞. 2,α α→∞ This together with the fact that r < r give α t 2,α r 2,α c c −c 0 lim 1 − = lim 1 − = (e ) = 1 α→∞ α→∞ t t r r 2,α 2,α and 2s (r −r )((1−2s)α/r ) 2 2 lim e = 0, α→∞ and (3.28) follows. By (3.28) and the fact that > 1 we get 2s r ,α 1 θ 1 θ 2 −r (log t ) α −r (log t ) α r ,α ϑ (α)  e t dt  e (t − c) r r ,α α! α! t −c r ,α 1 1 1 θ 1 2s 2s −r (log t ) α 2 R ·α 2 R ·α 2 r ,α 2 3 e t = e  e . r ,α α! α! This gives the result.   20 Page 28 of 34 E. Nabizadeh et al. 3.3 Proofs of main results Next we prove Proposition 3.9 and thereby Theorem 3.2. Proof of Proposition 3.9 It is clear that (2) ⇒ (3) ⇒ (4) ⇒ (5). By Proposition 3.13 it follows that (5) implies (1). We need to prove that (1) implies (2). Suppose (1) holds and let r = r. Choose r , r , r ∈ R such that r < r < r ≤ 4 1 2 3 0 1 2 2 ∞ d r < r and r r < r , χ ∈ R (C ), and let F be as in Lemma 3.14 with r and 3 4 1 4 α,χ 2 2 r ,r 2 3 r in place of r and r .If τ = 1, then Lemma 3.18 (2) gives 3 1 2 α − |c( F,α)|  r α! . (3.29) Let F (z) = c( F,α)ς z . (3.30) 0 α α∈N We claim that the series in (3.30) is uniformly convergent with respect to z in D (0). In fact, if |z |≤ r , then (3.14)gives j 4, j α d −2α α α d α |c( F,α)ς z |  α r r r =α ρ , 1 4 r r d 1, j 4, j d α where ρ ∈ R satisfies ρ = < 1. Since α ρ is convergent, Weier- j d + α∈N 2, j strass theorem shows that (3.30) is uniformly convergent and defines an analytic function in D (0). Hence, F ∈ A( D (0)). Furthermore, by the support of χ r 0 r 4, j 4, j we get ⎛ ⎞ ⎝ ⎠ ( F · χ) =  c( F,α)ς z · χ A 0 A α α∈N = c( F,α)ς  (z · χ) = c( F,α)e = F . (3.31) α A α d d α∈N α∈N Hence (2) holds, and the proof is complete. For future references we observe that if ς and χ are the same as in Lemma 3.14, then (3.30) shows that the relationship between c( F,α) and c( F ,α) is given by c( F ,α) = c( F,α)ς α! . (3.32) 0 α Proof of Theorem 3.4 The equivalence between (1) and (2) is clear. It is also obvious that (3) implies (2) in view of Proposition 3.9. We shall prove the equivalence between (1) and (3). d d Suppose (1) holds. Then (3.29) holds for every r ∈ R .Let r , R ∈ R be chosen + + ∞ d such that χ ∈ R (C ), F be as in Lemma 3.14 and let F be given by (3.30). By α,χ 0 r , R (3.29)wehave d −2α α |c( F,α)ς |  α r r 1 Paley–Wiener properties for spaces of entire functions Page 29 of 34 20 for every r ∈ R , giving that |c( F,α)ς |  r for every r ∈ R . This implies that the series in (3.30) is locally uniformly convergent d d with respect to z and defines an entire function on C . Hence F ∈ A(C ). Moreover, by (3.31) it follows that  ( F χ) = F, and we have proved that (1) implies (3). A 0 Next suppose that (3) holds. Then |c( F ,α)|  r α! for every r ∈ R .By(3.14) and (3.32) we get 1 1 −1 − α −1 α 2α − 2 2 |c( F,α)|=|c( F ,α)|ς α!  r ς  r R α! . α α Since r ∈ R can be chosen arbitrarily small we get α − |c( F,α)|  r α! d d for every r ∈ R . This implies that F ∈ A (C ). That is, we have proved that (3) 0 0, + 1 implies (1), and the result follows. Next we prove Proposition 3.10. Proof of Proposition 3.10 Suppose (3.3) holds, and let r , r ∈ R be such that r < 1 2 0 r < r < r. Then Lemma 3.18 gives 1 2 2τ +1 ·α 2r −τ |c( F,α)|  α! . 2τ + 1 By Lemma 3.14 and (3.32) we get 2τ +1 ·α 2r −τ d −2α |c( F ,α)|  α! α α!t 2τ + 1 2τ +1   2τ −1 ·α ·α 2 2 2r 2 R 2 2 −(τ −1) −2α −(τ −1) α! t = α! , 2τ + 1 2τ − 1 when 2τ +1 2τ −1 2τ − 1 2r − 2τ −1 R = t . 2 2τ + 1 This proves (1). 20 Page 30 of 34 E. Nabizadeh et al. Suppose instead that r , R , r , r ∈ R are such that (3.6) hold and r < r < 0 0 1 2 0 1 r < r, and that F ∈ A(C ) satisfies (3.4). Also let F =  ( F · χ). Then 2 0 A 0 2τ −1   2τ +1 2 2 2 R 2r 0 1 −2 < t . 2τ − 1 2τ + 1 By combining the latter estimate with Lemma 3.18 we get 2τ +1 ·α 2r −2α 1−τ |c( F ,α)|  t α! 2τ + 1 Hence Lemma 3.14 and (3.32)give 2τ +1 ·α 2r −τ |c( F,α)|  α! 2τ + 1 and Lemma 3.18 again implies that (3.3) holds with r in place of r . This gives the 2 0 result. 1 σ 1 Proof of Theorems 3.5–3.7 First suppose σ ∈ ( , 1), and let σ = and τ = . 2 2σ −1 2σ Then 2σ 2 2σ 2 = and = . σ + 1 2τ + 1 σ − 1 2τ − 1 Theorem 3.5 now follows from these observations and Proposition 3.10 in the case τ ∈ ( , 1). In the same way, Theorem 3.6 follows by choosing τ = 1 in Proposition 3.10. 1 σ 1 Finally, suppose σ ∈ (0, ), and let σ = and τ = . Then 2 1−2σ 2σ 2σ 2 2σ 2 = and = , σ + 1 2τ + 1 σ + 1 2τ − 1 and Theorem 3.7 follows from these observations and Proposition 3.10 in the case τ> 1. Next we prove Propositions 3.11 and 3.12 and thereby Theorem 3.8. Proof of Propositions 3.11 and 3.12 Let r , r , R ∈ R , j = 0, 1, 2, 3, be such that j j 1−2s 2s 1 − 2s r < r < r < r < r and R = s . 0 1 2 3 j First suppose that F ∈ A(C ) satisfies (3.7) and let F be the formal power series expansion with coefficients given by (3.32). Then F =  ( F · χ). A 0 Paley–Wiener properties for spaces of entire functions Page 31 of 34 20 By Lemma 3.19 we get 1 1 2s 2s −( R |α | +···+ R |α | ) 1,1 1 1,d d |c( F,α)|  e . Hence Lemma 3.14 and (3.32)give 1 1 2s 2s −2α −( R |α | +···+ R |α | ) 1,1 1 1,d d |c( F ,α)|  α!t e 1 1 2s 2s −( R |α | +···+ R |α | ) 2,1 1 2,d d e . In the last inequality we have used the fact that s < , which implies that R < R 2 1 and 1 1 2s 2s −2α ( R − R )|α | +···+( R − R )|α | 1,1 2,1 1 1,d 2,d d t α!  e . By applying Lemma 3.19 again it follows that F satisfies (3.8) with r in place of 0 3 r . This gives (1). Suppose instead that F ∈ A(C ) satisfies (3.8) and let F =  ( F · χ). Then 0 A 0 Lemma 3.19 gives 1 1 2s 2s −( R |α | +···+ R |α | ) 1,1 1 1,d d |c( F ,α)|  e , and it follows from Lemma 3.14 and (3.32) that 1 1 2s 2s −1 2α −( R |α | +···+ R |α | ) 1,1 1 1,d d |c( F,α)|  α! t e 1 1 2s 2s −( R |α | +···+ R |α | ) 1,1 1 1,d d e . By applying Lemma 3.19 again we deduce (3.7) with r in place of r , and Proposition 2 0 3.11 follows. Proposition 3.12 is a straight-forward consequence of (3.32). The details are left for the reader. 4 Characterizations of Pilipovic´ spaces In this section we combine Lemma 2.10 with Theorems 3.2–3.8 to get characterizations of Pilipovic´ spaces. We begin with the following characterization of H . The result is a straight-forward combination of Lemma 2.10 and Theorem 3.2. The details are left for the reader. Proposition 4.1 Let φ be as in (2.7),r ∈ R , χ be the characteristic function for D (0) and let s > 0. Then the following conditions are equivalent: (1) f ∈ H (R ); 1 20 Page 32 of 34 E. Nabizadeh et al. ∗  2d (2) f = V Ffor some F ∈ E (R ); φ s ∗  2d ∞ 2d (3) f = V Ffor some F ∈ E (R ) ∩ L (R ); ∗  2d ∞ 2d (4) f = V Ffor some F ∈ E (R ) ∩ L (R ) which satisfies −i x ,ξ F (x,ξ) = F (x − iξ)e χ(x,ξ) (4.1) for some r ∈ R , χ = χ and F ∈ A( D (0)). r 0 r ∞ d Remark 4.2 It is clear that χ in Proposition 4.1 can be chosen as any χ ∈ R (C ) with suitable support properties. 1 2 2 (|x | +|ξ | ) Remark 4.3 In (2.32) there is a factor e which is absent in (4.1). We notice ∞ d that this factor is not needed in (4.1) because R (C ) is invariant under multiplica- t ,t 1 2 tions of such functions. The next results follow from Lemma 2.10 and Theorems 3.4–3.8. The details are left for the reader. ∞ d Proposition 4.4 Let φ be as in (2.7) and χ ∈ R (C ). Then the following conditions are equivalent: (1) f ∈ H (R ); 0, ∗  2d ∞ 2d (2) f = V Ffor some F ∈ E (R ) ∩ L (R ) which satisfies (4.1) for some F ∈ A(C ). ∞ d 1 Proposition 4.5 Let φ be as in (2.7), χ ∈ R (C ), σ ∈ ( , 1) and let σ = . 2σ − 1 Then the following conditions are equivalent: d d (1) f ∈ H (R )( f ∈ H (R )); 0, σ σ ∗  2d ∞ 2d (2) f = V Ffor some F ∈ E (R ) ∩ L (R ) which satisfies (4.1) for some d  d F ∈ A (C )( F ∈ A (C )). 0 0 0, σ σ 0 0 ∞ d 1 Proposition 4.6 Let φ be as in (2.7), χ ∈ R (C ) and let σ = . Then the following conditions are equivalent: d d (1) f ∈ H (R )( f ∈ H (R )); 0, σ σ 2d ∞ 2d (2) f = V Ffor some F ∈ E (R ) ∩ L (R ) which satisfies (4.1) for some d d F ∈ A (C )( F ∈ A (C )). 0 0 0,1/2 0,1/2 Paley–Wiener properties for spaces of entire functions Page 33 of 34 20 ∞ d Proposition 4.7 Let φ be as in (2.7), χ ∈ R (C ), σ ∈ (0, ) and let σ = . 1 − 2σ Then the following conditions are equivalent: d d (1) f ∈ H (R )( f ∈ H (R )); 0, σ σ ∗  2d ∞ 2d (2) f = V Ffor some F ∈ E (R ) ∩ L (R ) which satisfies (4.1) for some d d F ∈ A (C )( F ∈ A (C )). 0  0 0, σ σ 0 0 ∞ d 1 Proposition 4.8 Let φ be as in (2.7), χ ∈ R (C ) and let s ∈ (0, ). Then the following conditions are equivalent: d d (1) f ∈ H (R )( f ∈ H (R )); s s ∗  2d ∞ 2d (2) f = V Ffor some F ∈ E (R ) ∩ L (R ) which satisfies (4.1) for some d d F ∈ A (C )( F ∈ A (C )). 0 s 0 0,s Acknowledgements Open access funding provided by Linnaeus University. Compliance with ethical standards Conflict of interest There are no conflict of interests. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. References 1. Bargmann, V.: On a Hilbert space of analytic functions and an associated integral transform. Commun. Pure Appl. Math. 14, 187–214 (1961) 2. Bargmann, V.: On a Hilbert space of analytic functions and an associated integral transform. Part II. A family of related function spaces. Application to distribution theory. Commun. Pure Appl. Math. 20, 1–101 (1967) 3. Cordero, E., Pilipovic, ´ S., Rodino, L., Teofanov, N.: Quasianalytic Gelfand–Shilov spaces with appli- cations to localization operators. Rocky Mt. J. Math. 40, 1123–1147 (2010) 4. Fernandez, C., Galbis, A., Toft, J.: The Bargmann transform and powers of harmonic oscillator on Gelfand–Shilov subspaces. RACSAM 111, 1–13 (2017) 5. Gröchenig, K.H.: Foundations of Time-Frequency Analysis. Birkhäuser, Boston (2001) 6. Hörmander, L.: The Analysis of Linear Partial Differential Operators, Vol. I–III. Springer-Verlag, Berlin Heidelberg NewYork Tokyo, 1983 (1985) 7. Janssen, A.M.E.M., Eijndhoven, S.J.L.: Spaces of type W, growth of Hermite coefficients, Wigner distribution, and Bargmann transform. J. Math. Anal. Appl. 152, 368–390 (1990) 8. Pilipovic, ´ S.: Generalization of Zemanian spaces of generalized functions which have orthonormal series expansions. SIAM J. Math. Anal. 17, 477–484 (1986) 9. Pilipovic, ´ S.: Tempered ultradistributions. Boll. U.M.I. 7, 235–251 (1988) 20 Page 34 of 34 E. Nabizadeh et al. 10. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. Academic Press, London New York (1979) 11. Toft, J.: The Bargmann transform on modulation and Gelfand–Shilov spaces, with applications to Toeplitz and pseudo-differential operators. J. Pseudo-Differ. Oper. Appl. 3, 145–227 (2012) 12. Toft, J.: Images of function and distribution spaces under the Bargmann transform. J. Pseudo-Differ. Oper. Appl. 8, 83–139 (2017) 13. Toft, J.: Paley-Wiener properties for spaces of power series expansions. Complex Var. Elliptic Equ. (2019). https://doi.org/10.1080/17476933.2019.1681414 Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

Paley–Wiener properties for spaces of entire functions

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Abstract

We deduce Paley–Wiener results in the Bargmann setting. At the same time we deduce characterisations of Pilipovic´ spaces of low orders. In particular we improve the characterisation of the Gröchenig test function space H = S , deduced in Toft (J Pseudo-Differ Oper Appl 8:83–139, 2017). Mathematics Subject Classification Primary 46F05 · 32A25 · 32A36; Secondary 35Q40 · 30Gxx 1 Introduction Paley–Wiener theorems characterize functions and distributions with certain restricted supports in terms of estimates of their Fourier–Laplace transforms. For example, let d d f be a distribution on R and let B (0) ⊆ R be the ball with center at origin and radius r . Then f is supported in B (0) if and only if 0 r N r | Im(ζ )| d | f (ζ )|  ζ  e ,ζ ∈ C , for some N ≥ 0. Furthermore, f is supported in B (0) and smooth, if and only if − N r | Im(ζ )| d | f (ζ )|  ζ  e ,ζ ∈ C , for every N ≥ 0. Joachim Toft joachim.toft@lnu.se Elmira Nabizadeh elmira.nabizadehmorsalfard@lnu.se Christine Pfeuffer christine.pfeuffer@mathematik.uni-halle.de Department of Mathematics, Linnæus University, Växjö, Sweden Department of Mathematics, Martin-Luther-University of Halle-Wittenberg, Halle (Saale), Germany 0123456789().: V,-vol 20 Page 2 of 34 E. Nabizadeh et al. A similar approach for ultra-regular functions of Gevrey types and corresponding ultra-distribution spaces can be done. In fact, let s > 1, D (R ) be the set of all Gevrey distributions of order s and let E (R ) be the set of all smooth functions with Gevrey regularity s. (See [6] and Sect. 2 for notations.) Then it can be proved that f ∈ D (R ) is supported in B (0), if and only if r | Im(ζ )|+r |ζ | d | f (ζ )|  e ,ζ ∈ C , for every r > 0. Furthermore f ∈ E (R ) is supported in B (0), if and only if s r r | Im(ζ )|−r |ζ | d | f (ζ )|  e ,ζ ∈ C , for some r > 0. We observe that s in the latter result can not be pushed to be smaller, because if s ≤ 1, it does not make any sense to discuss compact support properties of D (R ) and E (R ). In the paper we consider analogous Paley–Wiener properties after the Fourier– Laplace transform above is replaced by the reproducing kernel  for the Bargmann transform, and the image spaces are replaced by suitable subspaces of entire functions on C . These subspaces were considered in [4,12] and are given by d d d d A (C ) = A (C ), A (C ) = A (C ), r , 0, r , σ σ σ σ r >0 r >0 (1.1) d d d d A (C ) = A (C ), A (C ) = A (C ), s r ,s 0,s r ,s r >0 r >0 1 d d when σ> 0 and 0 < s < , where A (C ) and A (C ) are the sets of all entire r , r ,s F on C such that 2σ 1 σ +1 1−2s r |z| r (logz) | F (z)|  e respective | F (z)|  e . The spaces in (1.1) appear naturally when considering the Bargmann transform images of extended classes of Fourier invariant Gelfand–Shilov spaces, called Pilipovic´ spaces (see [4,12]). If (z,w) is the scalar product of z,w ∈ C , then the reproducing kernel of the Bargmann transform is given by −d (z, · )−| · | ( F )(z) = π  F , e when F is a suitable function or (ultra-)distribution. If R|z|−|z| 1 d z → F (z)e ∈ L (C ), R >0(1.2) Paley–Wiener properties for spaces of entire functions Page 3 of 34 20 holds and dλ(w) is the Lebesgue measure on C , then −d (z,w)−|w| ( F )(z) = π F (w)e dλ(w). A recent Paley–Wiener result with respect to the transform  and image spaces ∞ d  d ∞ d (1.1)isgiven in [12], where it is proved that if L (C ) = E (C ) ∩ L (C ), then d ∞ d d (E (C )) =  (L (C )) = A (C ). (1.3) A A c 1 ∞ d  d Evidently, L (C ) ⊆ E (C ), and the gap between these spaces are rather large. It might therefore be somewhat surprising that the first equality holds in (1.3). In Sect. 3 we improve (1.3) in different ways. Firstly we show that we may replace ∞ d ∞ d L (C ) in (1.3) with the smaller space L (C ) given by A,c ∞ d ∞ L (C ) = L (K ), A,c A,c where L (K ) is the set of all F · χ , where F is analytic in a neighbourhood of the A,c compact set K and χ is the characteristic function of K . Secondly, we may replace d  d E (C ) with the set E (C ) of all compactly supported Gevrey distributions of order s > 1. Summing up we improve (1.3)into d ∞ d d (E (C )) =  (L (C )) = A (C ), s > 1. (1.3) A A s A,c 1 In Sect. 3 we also deduce various kinds of related mapping properties when A (C ) ∞ d in (1.3) is replaced by any of the spaces in (1.1). More precisely, let χ ∈ L (C ) be non-negative, radial symmetric in each complex variable z and bounded from below by a positive constant near origin. Then we prove d d 1 (A (C ) · χ) = A (C ), σ ∈ ( , 1), σ = , A  0 0, σ σ 2 2σ − 1 d d 1 (A (C ) · χ) = A (C ), σ ∈ (0, ), σ = , A   0 σ σ 1 − 2σ d d (A (C ) · χ) = A (C ), s ∈[0, ), A s s and similarly for σ = and when A and A are replaced by A and A , s 0, 0,s σ σ respectively. (Cf. Theorems 3.2–3.8 and Propositions 3.9–3.11.) Finally, in Sect. 4 we use the results in Sect. 3 to deduce characterizations of Pilipovic´ spaces of small orders. We remark that some continuations of the present analysis, involving larger spaces of analytic functions or formal power series expan- sions, can be found in [13]. 20 Page 4 of 34 E. Nabizadeh et al. 2 Preliminaries In this section we recall some basic facts. We start by discussing Pilipovic´ spaces and some of their properties. Then we recall some facts on modulation spaces. Finally we recall the Bargmann transform and some of its mapping properties, and introduce suitable classes of entire functions on C . 2.1 The Pilipovic´ spaces In order to define Pilipovic´ spaces we recall the definition of Hermite functions. We recall that the Hermite function of order α ∈ N is defined by |x | d 1 2 − |α| |α| − α −|x | 4 2 2 h (x ) = π (−1) (2 α!) e (∂ e ). It follows that d 1 |x | − |α| − − 4 2 2 h (x ) = π (2 α!) e p (x ), α α for some polynomial p on R , which is called the Hermite polynomial of order α.The Hermite functions are eigenfunctions to the Fourier transform, and to the Harmonic oscillator H ≡|x | − which acts on functions and (ultra-)distributions defined on R . More precisely, we have H h = (2|α|+ d)h , H ≡|x | − . d α α d It is well-known that the set of Hermite functions is a basis for S (R ) and an 2 d 2 d orthonormal basis for L (R ) (cf. [10]). In particular, if f ∈ L (R ), then 2 2 f  = |c ( f ,α)| , 2 d h L (R ) α∈N where f (x ) = c ( f ,α)h (2.1) h α α∈N is the Hermite series expansion of f , and c ( f ,α) = ( f , h ) 2 d (2.2) h α L (R ) is the Hermite coefficient of f of order α ∈ R . In order to define the full scale of Pilipovic´ spaces, their order s should belong to the extended set R = R ∪{  ; σ ∈ R }, + σ + Paley–Wiener properties for spaces of entire functions Page 5 of 34 20 of R , with extended inequality relations as s < < s and  < 1 σ 2 σ σ 1 2 when s < ≤ s and σ <σ .(Cf.[12].) 1 2 1 2 For such s and r ∈ R we set 1 1 1 1 2s 2s −( ·α +···+ ·α ) ⎪ 1 r 1 r d ⎨ 1 d e , s ∈ R \{ }, ϑ (α) ≡ − (2.3) r ,s α 2σ r α! , s =  , ⎪ σ α 1 d r , s = ,α ∈ N and 1 1 1 2s 1 2s ( ·α +···+ ·α ) ⎪ 1 r 1 r d ⎨ 1 d e , s ∈ R \{ }, ϑ (α) ≡ (2.4) r ,s 2σ r α! , s =  , ⎪ σ α d r , s = ,α ∈ N . Definition 2.1 Let s ∈ R = R ∪{0}, and let ϑ and ϑ be as in (2.3) and (2.4). r ,s r ,s d  d (1) H (R ) consists of all finite Hermite series expansions (2.1), and H (R ) consists of all formal Hermite series expansions in (2.1); d d 2 d (2) if s ∈ R , then H (R ) (H (R )) consists of all f ∈ L (R ) such that s 0,s |c ( f , h )|  ϑ (α) h α r ,s d d holds true for some r ∈ R (for every r ∈ R ); + + d  d (3) if s ∈ R , then H (R ) (H (R )) consists of all formal Hermite series expan- 0,s sions in (2.1) such that |c ( f , h )|  ϑ (α) h α r ,s d d holds true for every r ∈ R (for some r ∈ R ). + + d d The spaces H (R ) and H (R ) are called Pilipovic ´ spaces of Roumieu respectively s 0,s d  d Beurling type of order s, and H (R ) and H (R ) are called Pilipovic ´ distribution s 0,s spaces of Roumieu respectively Beurling type of order s. d d Remark 2.2 Let S (R ) and (R ) be the Fourier invariant Gelfand–Shilov spaces of s s order s ∈ R and of Rourmeu and Beurling types respectively (see [12] for notations). Then it is proved in [8,9] that d d H (R ) = (R ) ={0}, s > , 0,s s d d H (R ) = (R ) ={0}, s ≤ , 0,s s 2 20 Page 6 of 34 E. Nabizadeh et al. d d H (R ) = S (R ) ={0}, s ≥ s s and d d H (R ) = S (R ) ={0}, s < . s s In Proposition 2.3 below we give further characterisations of Pilipovic´ spaces. Next we recall the topologies for Pilipovic´ spaces. Let s ∈ R , r > 0, and let f  and  f   be given by s;r H s;r f  ≡ sup |c ( f ,α)ϑ (α)|, s ∈ R , (2.5) H h r ,s s;r α∈N and f   ≡ sup |c ( f ,α)ϑ (α)|, s ∈ R . (2.6) H h r ,s s;r α∈N when f is a formal expansion in (2.1). Then H (R ) consists of all expansions s;r (2.1) such that  f  is finite, and H (R ) consists of all expansions (2.1) such s;r s;r d  d that  f   is finite. It follows that both H (R ) and H (R ) are Banach spaces s;r s;r s;r under the norms f → f  and f → f   , respectively. H H s;r s;r d d We let the topologies of H (R ) and H (R ) be the inductive respectively pro- s 0,s jective limit topology of H (R ) with respect to r > 0. In the same way, the s;r d  d topologies of H (R ) and H (R ) are the projective respectively inductive limit 0,s topology of H (R ) with respect to r > 0. It follows that all the spaces in Definition s;r d  d 2.1 are complete, and that H (R ) and H (R ) are Fréchet spaces with semi-norms 0,s f → f  and f → f  , respectively. s;r s;r The following characterisations for Pilipovic´ spaces can be found in [12]. The proof is therefore omitted. d d Proposition 2.3 Let s ∈ R ∪{0} and let f ∈ H (R ). Then f ∈ H (R )( f ∈ + 0,s d ∞ d H (R )), if and only if f ∈ C (R ) and satisfies H f  ∞ sup < ∞ N 2s h N ! N ∈N for every h > 0 (for some h > 0). From now on we let d |x | − − 4 2 φ(x ) = π e . (2.7) Paley–Wiener properties for spaces of entire functions Page 7 of 34 20 2.2 Spaces of entire functions and the Bargmann transform Let  ⊆ C be open. Then A() is the set of all analytic functions in . If instead ⊆ C is closed, then A() is the set of all functions which are analytic in an open neighbourhood of . In particular, if z ∈ C is fixed, then A({z }) is the set of all 0 0 complex-valued functions which are defined and analytic near z . We shall now consider the Bargmann transform which is defined by the formula d 1 1 − 2 4 2 (V f )(z) = π exp − (z, z+| y| ) + 2 z, y f ( y) dy, d 2 2 d 2 d when f ∈ L (R ) (cf. [1]). We note that for all f ∈ L (R ), then the Bargmann transform V f of f is the entire function on C , given by (V f )(z) = A (z, y) f ( y) dy, d d or (V f )(z) = f , A (z, · ), (2.8) d d where the Bargmann kernel A is given by d 1 1 − 2 4 2 A (z, y) = π exp − (z, z+| y| ) + 2 z, y . Here d d z,w= z w , when z = (z ,..., z ) ∈ C and w = (w ,...,w ) ∈ C , j j 1 d 1 d j =1 and otherwise · , · denotes the duality between test function spaces and their corresponding duals. We note that the right-hand side in (2.8) makes sense when d d f ∈ S (R ) and defines an element in A(C ), since y → A (z, y) can be inter- 1/2 d d preted as an element in S (R ) with values in A(C ). 1/2 2 d It was proved in [1] that f → V f is a bijective and isometric map from L (R ) 2 d 2 d d 2 d to the Hilbert space A (C ) ≡ B (C ) ∩ A(C ), where B (C ) consists of all mea- surable functions F on C such that F  2 ≡ | F (z)| dμ(z) < ∞. (2.9) −d −|z| d Here dμ(z) = π e dλ(z), where dλ(z) is the Lebesgue measure on C .We 2 d 2 d recall that A (C ) and B (C ) are Hilbert spaces, where the scalar products are given by 2 d ( F , G) ≡ F (z)G(z) dμ(z), F , G ∈ B (C ). (2.10) C 20 Page 8 of 34 E. Nabizadeh et al. 2 d If F , G ∈ A (C ), then we set  F  = F  and ( F , G) = ( F , G) . 2 2 2 2 A B A B Furthermore, Bargmann showed that there is a convenient reproducing formula on 2 d A (C ). More precisely, let (z,w) ( F )(z) ≡ F (w)e dμ(w), (2.11) R|z|−|z| 1 d when z → F (z)e belongs to L (C ) for every R ≥ 0. Here d d (z,w) = z w , when z = (z ,..., z ) ∈ C and w = (w ,...,w ) ∈ C , j j 1 d 1 d j =1 d d is the scalar product of z ∈ C and w ∈ C . Then it is proved in [1,2] that  is 2 d 2 d the orthogonal projection of B (C ) onto A (C ). In particular,  F = F when 2 d F ∈ A (C ). In [1] it is also proved that V h = e , where e (z) ≡ √ , z ∈ C . (2.12) d α α α α! 2 d In particular, the Bargmann transform maps the orthonormal basis {h } d in L (R ) α∈N 2 d bijectively into the orthonormal basis {e } d of monomials in A (C ). Hence, there α∈N is a natural way to identify formal Hermite series expansion by formal power series expansions F (z) = c( F,α)e (z), (2.13) α∈N by letting the series (2.1) be mapped into c ( f ,α)e (z). (2.14) h α α∈N 2 d 2 d It follows that if f , g ∈ L (R ) and F , G ∈ A (C ), then ( f , g) 2 d = c ( f ,α)c (g,α), h h L (R ) α∈N ( F , G) = c( F,α)c(G,α). (2.15) 2 d A (C ) α∈N Here and in what follows, ( · , · ) and ( · , · ) denote the scalar products in 2 d 2 d L (R ) A (C ) 2 d 2 d L (R ) and A (C ), respectively. Furthermore, c ( f ,α) = c( F,α) when F = V f . (2.16) h d We now recall the following spaces of power series expansions given in [12]. Paley–Wiener properties for spaces of entire functions Page 9 of 34 20 Definition 2.4 Let s ∈ R = R ∪{0}, and let ϑ and ϑ be as in (2.3) and (2.4). r ,s r ,s d d  d (1) A (C ) consists of all analytic polynomials on C , and A (C ) consists of all formal power series expansions on C in (2.13); d d 2 d (2) if s ∈ R , then A (C ) (A (C )) consists of all F ∈ L (C ) such that s 0,s |c( F , h )|  ϑ (α) α r ,s holds true for some r > 0 (for every r > 0); d  d (3) if s ∈ R , then A (C ) (A (C )) consists of all formal power series expansions s 0,s in (2.13) such that |c( F , h )|  ϑ (α) r ,s holds true for every r > 0(forsome r > 0). Let f ∈ H (R ) with formal Hermite series expansion (2.1). Then the Bargmann transform V f of f is defined to be the formal power series expansion (2.14). It 2 d follows that V agrees with the earlier definition when acting on L (R ), that V d d d  d is linear and bijective from H (R ) to A (C ), and restricts to bijections from the 0 0 spaces d d  d  d H (R ), H (R ), H (R ) and H (R ) (2.17) 0,s s s 0,s to d d  d  d A (C ), A (C ), A (C ) and A (C ) (2.18) 0,s s s 0,s respectively, when s ∈ R . We also let the topologies of the spaces in (2.18)be inherited from the spaces in (2.17). d  d d  d If s ∈ R , f ∈ H (R ), g ∈ H (R ), F ∈ A (C ) and G ∈ A (C ), then s s s s ( f , g) and ( F , G) are defined by the formula (2.15). It follows that (2.16) 2 d 2 d L (R ) A (C ) d d holds for such choices of f and F. Furthermore, the duals of H (R ) and A (C ) s s d  d can be identified with H (R ) and A (C ), respectively, through the forms in (2.15). s s The same holds true with H , H , A , and A 0,s 0,s 0,s 0,s in place of H , H , A , and A , s s s s respectively, at each occurrence. In order to identify the spaces of power series expansions above with spaces of analytic functions, we let 1 1 1−2s 1−2s ⎪ r (logz ) + ··· + r (logz ) , s < , 1 1 d d 2σ 2σ σ +1 σ +1 M (z) = 1,r ,s r |z | + ··· + r |z | , s =  ,σ > 0, 1 1 d d σ ⎪ 2 1 1 ⎩ |z| 1 s s − (r |z | + ··· + r |z | ), s ≥ , 1 1 d d 2 2 20 Page 10 of 34 E. Nabizadeh et al. M (z), s = , 1,r ,s 0 2 M (z) = 1,r ,s 2 2 r |z | + ··· + r |z | , s = , 1 1 d d 2σ 2σ σ −1 σ −1 r |z | + ··· + r |z | , s =  ,σ > 1, 1 1 d d σ M (z) = 2,r ,s 2 1 1 |z| 1 s s + (r |z | + ··· + r |z | ), s ≥ , 1 1 d d 2 2 M (z), s = , 2,r ,s M (z) = (2.19) 2,r ,s 1 2 2 r |z | + ··· + r |z | , s = , 1 1 d d when r ∈ R and z ∈ C . For conveniency we set M = M . r 1, ,r + 1 By [12] we have the following. The proof is therefore omitted. 0 0 Proposition 2.5 Let M ,M ,M and M be as in (2.19) when s ∈ R and 1,r ,s 2,r ,s 1,r ,s 2,r ,s r ∈ R . Then − M d d ∞ d d 1,r ,s A (C ) ={ F ∈ A(C ) ; Fe ∈ L (C ) for every r ∈ R }, s > 0, 0,s d d − M ∞ d d 1,r ,s A (C ) ={ F ∈ A(C ) ; Fe ∈ L (C ) for some r ∈ R }, s ≥ 0, d d − M ∞ d d 2,r ,s A (C ) ={ F ∈ A(C ) ; Fe ∈ L (C ) for every r ∈ R }, s > , s + d d − M ∞ d d 2,r ,s A (C ) ={ F ∈ A(C ) ; Fe ∈ L (C ) for some r ∈ R }, s > , 0,s + d d  d A (C ) = A(C ) and A (C ) = A({0}). 0, 1 1 Next we recall the link between the Bargmann transform and the short-time Fourier transform f → V f with window function φ given by (2.7), defined by −i· ,ξ V f (x,ξ) ≡ f , φ(·− x )e . Let S be the dilation operator given by 1 1 − − 2 2 (SF )(x,ξ) = F (2 x , −2 ξ), (2.20) 1 2d when F ∈ L (R ). Then it follows by straight-forward computations that loc d 1 2 2 1 1 (|x | +|ξ | ) −i x ,ξ 2 2 2 2 (V f )(z) = (V f )(x + iξ) = (2π) e e V f (2 x , −2 ξ) d d φ d 1 2 2 (|x | +|ξ | ) −i x ,ξ  −1 2 2 = (2π) e e (S (V f ))(x,ξ), (2.21) or equivalently, d 1 1 1 2 2 − − (|x | +|ξ | ) −i x ,ξ /2 − − 2 4 2 2 V f (x,ξ) = (2π) e e (V f )(2 x , −2 ξ) φ d d |·| − −i x ,ξ /2 − 2 2 = (2π) e S(e (V f ))(x,ξ) (2.22) We observe that (2.21) and (2.22) can be formulated as −1 V = U ◦ V , and U ◦ V = V , d V φ d φ V Paley–Wiener properties for spaces of entire functions Page 11 of 34 20 2d  d where U is the linear, continuous and bijective operator on D (R )  D (C ), given by d 1 1 1 2 2 (|x | +|ξ | ) −i x ,ξ 2 2 2 2 (U F )(x,ξ) = (2π) e e F (2 x , −2 ξ). (2.23) Let D (z ) be the polydisc d,r 0 { z = (z ,..., z ) ∈ C ;|z − z | < r , j = 1,..., d }, 1 d j 0, j j with center and radii given by d d z = (z ,..., z ) ∈ C and r = (r ,..., r ) ∈ (0, ∞) . 0 0,1 0,d 1 d Then A(C ) = A( D (z)), A({0}) = A( D (z )). d,r d,r 0 d d r ∈(0,∞) r ∈(0,∞) 2.3 Hilbert spaces of power series expansions and analytic functions The spaces in Definition 2.4 can also be described by related unions and intersections of Hilbert spaces of analytic functions and power series expansions as follows. (See also [12].) d d Let ϑ be a weight on N , ω be a weight on C , and let ⎛ ⎞ 1 ⎝ ⎠ F  ≡ |c( F , α)ϑ (α)| (2.24) 2 d A (C ) [ϑ ] α∈N when F ∈ A (C ) is given by (2.13), and F  ≡ | F (z)ω(2 z)| dμ(z) (2.25) 2 d A (C ) (ω) d 2 d  d when F ∈ A(C ).Welet A (C ) be the set of all F ∈ A (C ) such that  F  is [ϑ ] 0 A [ϑ ] 2 d d finite, and A (C ) be the set of all F ∈ A(C ) such that  F  2 is finite. It follows (ω) A (ω) that these spaces are Hilbert spaces under these norms. If ϑ and ω are related to each others as 2 α ϑ(α) = ω (r ) r dr (2.26) α! and |z| 2 2 ω(z) = e ω (|z | ,..., |z | ), (2.27) 0 1 d 20 Page 12 of 34 E. Nabizadeh et al. for some suitable weight ω on R , then the following multi-dimensional version 2 d 2 d of [7, Theorem (4.1)] shows that A (C ) = A (C ) with equal norms. Here we [ϑ ] (ω) identify entire functions with their power series expansions at origin. Consequently, 2 d 2 d the Bargmann transform is bijective and isometric from H (R ) to A (C ) for such [ϑ ] (ω) choices of ϑ and ω. We omit the proof since the result is an immediate consequence of [12, Theorem 3.5]. d d Theorem 2.6 Let α ∈ N and ω be a positive measurable function on R .Alsolet d d ϑ and ω be weights on N and C , respectively, related to each others by (2.26) and (2.27), and such that |α| ϑ(α), α ∈ N , (2.28) (α!) 2 d 2 d holds for every r > 0. Then A (C ) = A (C ) with equality in norms. [ϑ ] (ω) In our situation, the involved weights should satisfy a split condition. In one dimen- sion, (2.26), (2.27) and (2.28) take the forms 2 α ϑ (α ) = ω (r ) r dr ,α ∈ N, (2.26) j j 0, j j α ! j R |z | ω (z ) = e ω (|z | ), z ∈ C (2.27) j j 0, j j j and |α | ϑ (α ), r > 0,α ∈ N. (2.28) j j j (α !) Lemma 2.7 Let ω be weights on R , ω be weights on C and ϑ be weights on 0, j + j j N such that (2.26) –(2.28) hold, j = 1,..., d , and set ω (z) ≡ ω (z ), 0 0, j j j =1 z ≡ (z ,..., z ) ∈ C .If ϑ and ω are given by (2.26), and (2.27), then 1 d ϑ(α) = ϑ (α ), α = (α ,...,α ) ∈ N (2.29) j j 1 d j =1 and ω(z) = ω (z ), z = (z ,..., z ) ∈ C . (2.30) j j 1 d j =1 Paley–Wiener properties for spaces of entire functions Page 13 of 34 20 α 1 Proof By [12, Theorem 3.5] and its proof, it follows that ω · r ∈ L (R ) for all 0, j + j ∈{1,..., d} and α ∈ N. Hence, Fubini’s theorem gives ⎛ ⎞ 1 2 d 1 1 2 α 2 ⎝ ⎠ ϑ(α) = ω (r ) r dr = ω (r ) r dr 0 0, j j j α! α! R 0 j =1 = ϑ (α ), j j j =1 and (2.29) follows. The assertion (2.30) follows from the definitions. 2.4 A test function space introduced by Gröchenig In this section we recall some comparison results deduced in [12], between a test function space, S (R ), introduced by Gröchenig in [5] to handle modulation spaces with elements in spaces of ultra-distributions. The definition of S (R ) is given as follows. d |x | − − d d 4 2 Definition 2.8 Let φ(x ) = π e . Then S (R ) and S (R ) consist of all f ∈ C G d ∗ ∞ d  d  d S (R ) such that f = V F,for some F ∈ L (R ) ∩ E (R ) and F ∈ E (R ), respectively. It follows that f ∈ S (R ), if and only if d 1 2 − − |x − y| i x ,η 2 2 f (x ) = (2π) F ( y,η)e e dydη, (2.31) 2d ∞ d  d for some F ∈ L (R ) ∩ E (R ). Remark 2.9 By the identity (V h, F ) = (h, V F ) and the fact that the map ( f ,φ) → d d 2d ∗ V f is continuous from S (R ) × S (R ) to S (R ), it follows that f = V F is d  2d uniquely defined as an element in S (R ) when F ∈ S (R ) (cf. [3]). In particular, d d the space S (R ) in Definition 2.8 is well-defined, and it is evident that S (R ) ⊆ G C S (R ). The following is a restatement of [12, Lemma 4.9]. The result is essential when deducing the characterizations of Pilipovic´ spaces in Sect. 4. ∞ d  d ∗ Lemma 2.10 Let F ∈ L (C ) ∪ E (C ). Then the Bargmann transform of f = V F is given by  F , where A 0 √ √ d 1 2 2 3 (|x | +|ξ | ) −i x ,ξ 4 2 F (x + iξ) = (2π ) F ( 2x , − 2ξ)e e . (2.32) d d Moreover, the images of S (R ) and S (R ) under the Bargmann transform are C G given by ∞ d  d  d {  F ; F ∈ L (C ) ∩ E (C ) } and {  F ; F ∈ E (C ) }, (2.33) A A respectively. 20 Page 14 of 34 E. Nabizadeh et al. The next results follow from [12, Theorem 4.10]. The proof is therefore omitted. d d d Proposition 2.11 It holds S (R ) = S (R ) = H (R ). C G Due to the image properties for the spaces in Proposition 2.11 under the Bargmann transform, the next result is equivalent with the previous one. Proposition 2.12 The sets in (2.33) are equal to A (C ). In the next section we extend Propositions 2.11 and 2.12 by proving that the conclu- sions in Proposition 2.12 hold for suitable smaller and larger sets than those in (2.33). We also deduce similar identifications for other Pilipovic´ spaces and their Bargmann images. 3 Paley–Wiener properties for Bargmann–Pilipovic´ spaces In this section we consider spaces of compactly supported functions with interiors in d  d A (C ) or in A (C ). We show that the images of such functions under the reproducing kernel  are equal to A (C ), for some other choice of s ≤  . In the first part we state A s 1 the main results given in Theorems 3.2–3.4. They are straight-forward consequences of Propositions 3.9, where more detailed information concerning involved constants are given. Thereafter we deduce results which are needed for their proofs. Depending of the choice of s, there are several different situations for characterizing A (C ). This gives rise to a quite large flora of main results, where each one takes care of one situation. 2 2 First we note that if A = A , then a split of the variables in the weight ω in [ϑ ] (ω) 2 2 A induce a split of the variables in ϑ in A . (See Lemma 2.7.) (ω) [ϑ ] In order to present the main results, it is suitable to make the following definition. ∞ d Definition 3.1 Let t , t ∈ R be such that t ≤ t . Then the function χ ∈ L (C ) is 1 2 1 2 called positive, bounded and radial symmetric with respect to t and t , if the following 1 2 conditions are fulfilled: ∞ d • χ ∈ L (C ) ∩ E ( D (0)) is non-negative; • χ(z ,..., z ) = χ (|z |,..., |z |) for some function χ ; 1 d 0 1 d 0 • χ ≥ c on D (0) for some constant c > 0. The set of positive, bounded and radial symmetric functions with respect to t and t 1 2 ∞ d ∞ d is denoted by R (C ), and R (C ) is defined by t ,t 1 2 ∞ d ∞ d R (C ) ≡ R (C ). t ,t 1 2 t ≤t ∈R 1 2 3.1 Main results We begin with characterizing the largest spaces in our investigations, which appears when s =  , and then proceed with spaces of decreasing order. First we recall that 1 Paley–Wiener properties for spaces of entire functions Page 15 of 34 20 d d elements in A (C ) and A (C ) fulfill conditions of the forms s 0,s 2σ 2σ σ +1 σ +1 r |z | +···+r |z | 1 1 d d | F (z)|  e (3.1) when s =  and 1 1 1−2s 1−2s r (logz ) +···+r (logz ) 1 1 d d | F (z)|  e (3.2) when s ∈[0, ). Theorem 3.2 Let F ∈ A(C ), σ = 1 and s > 1. Then the following conditions are equivalent: (1) F ∈ A (C ); (2) (3.1) holds for some r ∈ R ; d d ∞ d (3) For some r ∈ R and every r ∈ R with r ≤ r and every χ ∈ R (C ), there 0 0 + + r ,r exists F ∈ A( D (0)) such that F =  ( F · χ); 0 r A 0 d d ∞ d (4) For some r ∈ R and every r ∈ R with r < r and some χ ∈ R (C ), there 0 0 + + r ,r exists F ∈ A( D (0)) such that F =  ( F · χ); 0 r A 0 d ∞ d (5) There exists F ∈ E (C ) ∩ L (C ) such that F =  F ; 0 A 0 (6) There exists F ∈ E (C ) such that F =  F . 0 A 0 Remark 3.3 Since d ∞ d  d  d E (C ) ∩ L (C ) ⊆ E (C ) ⊆ E (C ), Theorem 3.2 still holds true after E has been replaced by E in (6). d ∞ d Theorem 3.4 Let F ∈ A(C ), σ = 1 and χ ∈ R (C ). Then the following is true: (i) The following conditions are equivalent: (1) F ∈ A (C ); 0, (2) (3.1) holds for every r ∈ R ; (3) There exists F ∈ A(C ) such that F =  ( F · χ); 0 A 0 d d (ii) The map F →  ( F · χ) from A(C ) to A (C ) is a homeomorphism. A 0, The next result deals with the case when s =  with σ ∈ ( , 1). d ∞ d Theorem 3.5 Let F ∈ A(C ), χ ∈ R (C ), σ ∈ ( , 1) and let σ = . 2σ − 1 Then the following is true: (i) The following conditions are equivalent: 20 Page 16 of 34 E. Nabizadeh et al. d d (1) F ∈ A (C ) (F ∈ A (C )); 0, σ σ (2) (3.1) holds for some (for every) r ∈ R ; d d (3) There exists F ∈ A (C ) (F ∈ A (C )) such that F =  ( F · χ); 0 0 A 0 0, σ σ 0 0 d d  d (ii) The mappings F →  ( F · χ) from A (C ) to A (C ) and from A (C ) 0, σ σ σ 0 0 to A (C ) are homeomorphisms. 0, The next result deals with the case when s =  with σ = . d ∞ d Theorem 3.6 Let F ∈ A(C ), σ = and χ ∈ R (C ). Then the following is true: (i) The following conditions are equivalent: d d (1) F ∈ A (C ) (F ∈ A (C )); 0, σ σ (2) (3.1) holds for some (for every) r ∈ R ; d d (3) There exists F ∈ A (C ) (F ∈ A (C )) such that F =  ( F · χ); 0 0 0,1/2 A 0 0,1/2 d d (ii) The mappings F →  ( F · χ) from A (C ) to A (C ) and from 0,1/2 1/2 d d A (C ) to A (C ) are homeomorphisms. 0,1/2 0, 1/2 The next result deals with the case when s =  with σ ∈ (0, ). d ∞ d 1 Theorem 3.7 Let F ∈ A(C ), χ ∈ R (C ), σ ∈ (0, ) and let σ = . 1 − 2σ Then the following is true: (i) The following conditions are equivalent: d d (1) F ∈ A (C ) (F ∈ A (C )); 0, σ σ (2) (3.1) holds for some (for every) r ∈ R ; d d (3) There exists F ∈ A (C ) (F ∈ A (C )) such that F =  ( F · χ); 0  0 0, A 0 σ σ 0 0 d d d (ii) The mappings F →  ( F · χ) from A (C ) to A (C ) and from A (C ) A   0, σ σ σ 0 0 to A (C ) are homeomorphisms. 0, In the next result we consider the case when s ∈[0, ) is real. d ∞ d 1 Theorem 3.8 Let F ∈ A(C ), χ ∈ R (C ),s ∈[0, ). Then the following is true: (i) The following conditions are equivalent: d d (1) F ∈ A (C ) (F ∈ A (C )); s 0,s (2) (3.2) holds for some (for every) r ∈ R ; d d (3) There exists F ∈ A (C ) (F ∈ A (C )) such that F =  ( F · χ); 0 s 0 0,s A 0 d d d (ii) The mappings F →  ( F · χ) from A (C ) to A (C ) and from A (C ) to A s s 0,s A (C ) are homeomorphisms. 0,s Paley–Wiener properties for spaces of entire functions Page 17 of 34 20 The previous theorems are essentially consequences of Propositions 3.9–3.11, where more detailed information about involved constants are given. d d Proposition 3.9 Let F ∈ A(C ), σ = 1,s > 1 and r ∈ R . Then the following conditions are equivalent: (1) For some r ∈ R such that r < r, (3.1) holds with r in place of r ; 0 0 0 d d (2) For some t ∈ R such that t < r , every t ∈ R with t ≤ t < r and every 1 1 2 1 2 + + ∞ d χ ∈ R (C ), there exists F ∈ A( D (0)) such that F =  ( F · χ); 0 r A 0 t ,t 1 2 (3) For some F ∈ A( D (0)) it holds F =  ( F · χ ); 0 r A 0 D (0) ∞ d (4) For some F ∈ E ( D (0)) ∩ L (C ) it holds F =  F ; 0 r A 0 (5) For some r ∈ R such that r < r , there exists F ∈ E ( D (0)) such that 0 0 0 r + s 0 F =  F . A 0 Theorems 3.5–3.7 essentially follow from the following proposition. 1 d Proposition 3.10 Let τ> ,r , t , t ∈ R be such that t ≤ t , and let χ ∈ 1 2 1 2 ∞ d R (C ). Then the following is true: t ,t 1 2 (1) Let F ∈ A(C ) be such that 2 2 2τ +1 2τ +1 r |z | +···+r |z | 0,1 1 0,d d | F (z)|  e (3.3) d d holds for some r ∈ R such that r < r . Then for some r ∈ R such that 0 0 0 + + r < r , there exists F ∈ A(C ) such that F =  ( F · χ) and 0 0 A 0 2 2 2τ −1 2τ −1 R |z | +···+ R |z | 0,1 1 0,d d | F (z)|  e , (3.4) where 2τ +1 2τ −1 2τ − 1 2r − 2τ −1 R = t ; (3.5) 2 2τ + 1 (2) Let r , R ∈ R be such that r < r and 0 0 0 2τ +1 2τ −1 2τ − 1 2r − 2τ −1 R = t , (3.6) 2 2τ + 1 d d F ∈ A(C ) be such that (3.4) hold and let F =  ( F · χ). Then F ∈ A(C ) 0 A 0 and satisfies (3.3) for some r ∈ R such that r < r. 0 0 Theorem 3.8 follows from the following two propositions, where the first one concerns the case when s > 0 and the second one make a more detailed explanation of the case s = 0, i. e. the case of analytic polynomials. 1 d ∞ d Proposition 3.11 Let s ∈ (0, ),r ∈ R , and let χ ∈ R (C ). Then the following is true: 20 Page 18 of 34 E. Nabizadeh et al. (1) Suppose F ∈ A(C ) satisfies 1 1 1−2s 1−2s r (logz ) +···+r (logz ) 0,1 1 0,d d | F (z)|  e (3.7) for some r ∈ R such that r < r . Then there is an F ∈ A(C ) such that 0 0 0 F =  ( F · χ) and A 0 1 1 1−2s 1−2s r (logz ) +···+r (logz ) 0,1 1 0,d d | F (z)|  e (3.8) for some r ∈ R such that r < r; 0 0 (2) Suppose F ∈ A(C ) satisfies (3.8) for some r ∈ R such that r < r , and let 0 0 0 F =  ( F · χ). Then F ∈ A(C ) and satisfies (3.7) for some r ∈ R such A 0 0 that r < r. ∞ d Proposition 3.12 Let χ ∈ R (C ) and let N ≥ 0 be an integer. Then the following is true: (1) Suppose F ∈ A(C ) is given by F (z) = c( F,α)z , (3.9) |α|≤ N where {c( F,α)} ⊆ C. Then there is an F ∈ A(C ) such that F =  ( F · |α|≤ N 0 A 0 χ) and F (z) = c( F ,α)z , (3.10) 0 0 |α|≤ N where {c( F ,α)} ⊆ C and satisfies c( F ,α) = 0 when c( F,α) = 0; 0 |α|≤ N 0 (2) Suppose F ∈ A(C ) satisfies (3.10) for some {c( F ,α)} ⊆ C, and let F = 0 0 |α|≤ N ( F · χ). Then F ∈ A(C ) and satisfies (3.9) for some {c( F,α)} ⊆ C A 0 |α|≤ N such that c( F,α) = 0 when c( F ,α) = 0. 3.2 Preparing results and their proofs For the proofs of Propositions 3.9–3.11 and thereby of Theorems 3.2–3.8 we need some preparatory results. Because the proof of Proposition 3.9 needs some room, we put parts of the statement in the following separate proposition. At the same time we slightly refine some parts concerning the image of compactly supported elements in L under the map  . Proposition 3.13 Let s > 1,r , r ∈ R be such that r < r and suppose that either 0 0 F ∈ E ( D (0)), F ∈ E ( D (0)), F ∈ E ( D (0)) or F ∈ L ( D (0)). 0 r 0 r 0 r 0 r s 0 0,s 0 0 Then F =  F ∈ A(C ) and satisfies A 0 r |z |+···+r |z | 1 1 d d | F (z)|  e . Paley–Wiener properties for spaces of entire functions Page 19 of 34 20 Proof By the inclusions E → E → E → E 0,s+ε s 0,s when ε> 0, it suffices to consider the case when F ∈ E ( D (0)) or F ∈ L ( D (0)) 0 r 0 r s 0 hold. Let r = r. First suppose that F ∈ E ( D (0)) holds, choose r ∈ R such that 2 0 r 1 s 0 + 2 2 −(| y| +|η| ) (z, y+i η) d r < r < r , ( y,η) = e and let  ( y,η) = e . By identifying C 0 1 2 z 2d with R and using the fact that F ∈ E ( D (0)) we obtain 0 r s 0 D ( ) z L ( D (0)) −d 1 | F (z)|= π | F , |  sup (3.11) A 0 0 z |α| 2d h α! α∈N 2d 2d for every h > 0. We also have  ∈ E (R )→ E (R ), which implies 1 1/2 0,s |α| α s D  ∞  h α! L ( D (0)) for every h > 0. Furthermore, α (z, y+i η) r |z |+···+r |z | 1,1 1 1,d d | D  ( y,η)|=|m (z)e |≤|m (z)|e , y + i η ∈ D (0), z α α r where α +α j d+ j d 2d m (z) = z , z ∈ C ,α ∈ N . j =1 By choosing h = 4 and h = 1 above, and letting y + i η ∈ D (0), Leibnitz rule 1 2 r gives −|α| −s α 4 α! | D ( )( y,η)| −|α| −s γ α−γ ≤ 4 α! | D  ( y,η)|| D ( y,η)| γ ≤α ⎛ ⎞ −|α| −s s r |z |+···+r |z | 1,1 1 1,d d ⎝ ⎠ 4 α! |m (z)|(α − γ)! e γ ≤α ⎛ ⎞ −|α| −s r |z |+···+r |z | 1,1 1 1,d d ⎝ ⎠ 4 |m (z)|γ ! e γ ≤α ⎛ ⎞ γ γ j d+ j |z | |z | j j r |z |+···+r |z | 1,1 1 1,d d ⎝ ⎠ ≤ sup · e . (3.12) s s γ ! γ ! γ ≤α j d+ j j =1 20 Page 20 of 34 E. Nabizadeh et al. In the last inequality we have used that the number of terms in the sums are bounded |α| k by 2 , and that ≤ 2 when n, k are non-negative integers such that k ≤ n. By combining (3.12) with the estimate γ γ j j 1 |z | (|z | ) j j s s|z | = ≤ e , γ ! γ ! j j we get ∞ 1 1 D ( ) z L ( D (0)) r s s 1 2s(|z | +···+|z | ) r |z |+···+r |z | 1 d 1,1 1 1,d d sup  e e |α| 2d h α! α∈N r |z |+···+r |z | 2,1 1 2,d d e . In the last inequality we have used the fact that r < r and s > 1. From the latter 1 2 estimate and (3.11) we obtain r |z |+···+r |z | 2,1 1 2,d d | F (z)|  e , A 0 and the result follows when F ∈ E ( D (0)). 0 r s 0 ∞ d Suppose instead that F ∈ L ( Q) holds, where Q = D (0) ⊆ C , and let 0 r Q = D (0) ⊆ C. Then j r 2, j (z,w) | F (z)|  | F (z)||e | dλ(w) A 0 0 |z ||w | r |z |+···+r |z | j j 2,1 1 2,d d ≤ F  ∞ e dλ(w )  e , 0 L j j =1 and the result follows in this case as well. In the next lemma we give options on compactly supported functions which are mapped on the basic monomials, e by the operator  . α A d ∞ d Lemma 3.14 Let t , t ∈ R be such that t ≤ t , χ ∈ R (C ), and let χ be such 1 2 1 2 0 + t ,t 1 2 that χ (|z |,..., |z |) = χ(z ,..., z ).If 0 1 d 1 d F (z) = ς z χ(z), α,χ α with −1 1 2 −d −|u| 2α d ς = 2 α! χ (u)e u u ··· u du , ={ u ∈ R ; u ≤ t }, α 0 1 d t (3.13) then the following is true: Paley–Wiener properties for spaces of entire functions Page 21 of 34 20 (1)  F = e ; A α,χ α (2) for some constant C > 0 which only depends on χ  ∞ , c in Definition 3.1 and the dimension d , it holds ⎛ ⎞ ⎛ ⎞ d d 1 2 1 −1 −2 −2α |t | −2 −2α ⎝ ⎠ ⎝ ⎠ 2 2 C t (α + 1) t α! ≤ ς ≤ Ce t (α + 1) t α! . j α j 2, j 2 1, j 1 j =1 j =1 (3.14) Proof By using polar coordinates in each complex variable when integrating we get −d α (z,w)−|w| ( F )(z) = π ς w χ(w)e dλ(w) A α,χ α −d α −|u| = π ς I (u, z)u χ (u)e u ··· u du, (3.15) α α 0 1 d where ⎛ ⎞ d d −i θ i α,θ  z u e j j ⎝ ⎠ I (u, z) = e e dθ = I (u , z ) (3.16) α α j j [0,2π) j =1 j =1 with 2π −i θ i α θ z u e j j j j I (u , z ) = e e dθ . α j j j By Taylor expansions we get ∞ k k −ikθ 2π j z u e j j i α θ j j I (u , z ) = e dθ α j j j k! k=0 α α j j ∞ k k 2π z u 2π z u j j j j i (α −k)θ j j = e dθ = k! α ! k=0 By inserting this into (3.15) and (3.16) we get α α u z 2 −d d α −|u| ( F )(z) = π ς (2π) u χ (u)e u ··· u du A α,χ α 0 1 d α! 1 2 d − 2α −|u| = 2 ς α! u χ (u)e u ··· u du e (z) = e (z) α 0 1 d α α and (1) follows. 20 Page 22 of 34 E. Nabizadeh et al. Since χ is non-negative and fullfills χ ≥ c on D (0) we get −1 2 1 |t | 2α ς  e α! u u ··· u du α 1 2 ⎛ ⎞ ⎛ ⎞ −1 2α +2 d d 2 1 2 1 1, j |t | |t | −2 −2α 2 2 ⎝ ⎠ ⎝ ⎠ 2 2 = e α!  e α! (t (α + 1)) t 1, j 1 2α + 2 j =1 j =1 which gives the right inequality in (3.14). By the support properties of χ we also have −1 2α ς  α! t u ··· u du , α 1 2 and the left inequality in (3.14) follows by similar arguments, and (2) follows. The next lemma shows that we may estimate entire functions by different Lebesgue norms. We omit the proof, since the result follows from [11, Theorem 3.2]. Lemma 3.15 Suppose s,τ ∈ R and r , r ∈ R are such that 1 1 s < ,τ > − and r < r . 2 2 Let p, q ∈[1, ∞],F ∈ A(C ) and set 2 2 2τ +1 2τ +1 M (z) = r |z | + ··· + r |z | 1,r 1 1 d d and 1 1 1−2s 1−2s M (z) = r (logz ) + ··· + r (logz ) . 2,r 1 1 d d Then − M − M j ,r j ,r F · e  p d   F · e  q d . L (C ) L (C ) The next lemma relates Lebesgue estimates of entire functions with estimates on corresponding Taylor coefficients. Here we let the Gamma function on R be defined by (x ,..., x ) = (x ), d 1 d j j =1 where  is the Gamma function on R . + Paley–Wiener properties for spaces of entire functions Page 23 of 34 20 1 d Lemma 3.16 Let τ> − ,r ∈ R ,M be the same as in Lemma 3.15, ω(z) = 1,r 1 2 |z| − M (z) d d 1,r e ,z ∈ C , and let α = (1,..., 1) ∈ N .Alsolet ((2τ + 1)(α + α )) −(2τ +1)(α+α ) ϑ (α) = (2τ + 1)(2r ) . α! If F ∈ A(C ) is given by (2.13), then ⎛ ⎞ 1 − M 2 1,r ⎝ ⎠ F · e  2 d = π |c( F,α)ϑ (α)| , L (C ) α∈N 2 d 2 d and A (C ) = A (C ) with equality in norms. [ϑ ] (ω) Proof Since 2τ +1 − M (z) −r |z | 1,r j j e = e , j =1 Lemma 2.7 shows that we may assume that d = 1, giving that r = r and α = 1. 1 0 2 d 2 d In view of Theorem 2.6 we have A (C ) = A (C ) with equality in norms, [ϑ ] (ω ) when 2τ +1 −2rt α ϑ(α) = e t dt . α! 2τ +1 By u = 2rt as new variables of integration we obtain −(2τ +1)(α+1) −u α(2τ +1)+2τ ϑ(α) = (2τ + 1)(2r ) e u du α! ((2τ + 1)(α + 1)) −(2τ +1)(α+1) = (2τ + 1)(2r ) , α! which implies that θ = θ , and the result follows. 1,r We also need the following version of Stirling’s formula. Lemma 3.17 Let α ≥ 0 be an integer and let τ> − . Then ((2τ + 1)(α + 1)) (2τ +1)·α τ 2τ (2τ + 1) (α + 1) · α! . (3.17) α! 20 Page 24 of 34 E. Nabizadeh et al. Lemma 3.17 follows by repeated applications of Stirling’s formula and the standard limit lim 1 + = e t →∞ for every x ∈ R. In order to be self-contained we present the arguments. τ τ Proof The result is obviously true for α = 0. For α ≥ 1wehave α  (α + 1) .A combination of the the latter relations and Stirling’s formula gives (2τ +1)·α+2τ + α ((2τ + 1)(α + 1)) ((2τ + 1)α + 2τ ) e α+2τ(α+1) α+ α! e 2 (2τ +1)·α+2τ + 1 2τ (2τ +1)·α+2τ + 2τ(α+1) (2τ + 1) 1 + α (2τ + 1)α 2τ(α+1) 2τ(α+1) (2τ +1)·α (2τ +1)·α τ 2τ (2τ + 1)  (2τ + 1) (α + 1) α! , (3.18) 2τα and the result follows. Proposition 3.9 essentially follows from the following lemma. Lemma 3.18 Let τ> − and r , r ∈ R be such that r < r . Then the following is 0 0 true: (1) If 2τ +1 ·α 2r −τ |c( F,α)|  α! , (3.19) 2τ + 1 then 2 2 2τ +1 2τ +1 r |z | +···+r |z | 1 1 d d | F (z)|  e ; (3.20) (2) If 2 2 2τ +1 2τ +1 r |z | +···+r |z | 0,1 1 0,d d | F (z)|  e (3.21) then 2τ +1 ·α 2r −τ |c( F,α)|  α! (3.22) 2τ + 1 Proof Let ϑ be the same as in Lemma 3.16 and r ∈ R be such that r < r < r. r 1 0 1 First we prove (1). Suppose that (3.19) holds and let α = (1,..., 1) ∈ N .Alsolet M be the same as in Lemma 3.15. Then Lemmas 3.15 and 3.16 give 1,r ⎛ ⎞ − M − M 1,r 1,r ⎝ ⎠ F · e  ∞ d   F · e  2 d  c( F,α)ϑ (α) L (C ) L (C ) 1 α∈N Paley–Wiener properties for spaces of entire functions Page 25 of 34 20 ⎛ ⎞ 2τ +1 2 (α+α ) ⎝ ⎠ = c( F,α)ϑ (α) α∈N sup c( F,α)ϑ (α) (α + 1) r j α∈N j =1 2τ +1 ·α 2τ + 1 sup |c( F,α)| α! . 2r d 0 α∈N Here the second inequality follows from the fact that ⎛ ⎛ ⎞ ⎞ (2τ +1)·(α+α ) τ r ⎝ ⎝ ⎠ ⎠ (α + 1) j =1 α∈N is convergent since r < r , and the fifth relation follows from Lemma 3.17.This 0 1 implies that (3.20) holds and (1) follows. Next assume that (3.21) holds. By Lemma 3.15 we get − M − M 1,r 1,r 0 1 F · e  ∞ d   F · e  2 d L (C ) L (C ) ⎛ ⎞ 1 ⎝   ⎠ c( F,α)ϑ (α) ≥ sup |c( F,α)| ϑ (α) r r 1 1 α∈N α∈N ⎛ ⎞ 2τ +1 2τ + 1 τ ⎝ ⎠ sup |c( F,α)| (α + 1) α! 2r α∈N j =1 2τ +1 2τ + 1 sup |c( F,α)| α! , 2r α∈N where the third inequality follows from Lemma 3.17. This gives (2). Proposition 3.11 mainly follows from the following result. d 1 d Lemma 3.19 Let r , r ∈ R be such that r < r, s ∈ (0, ) and let F ∈ A(C ). Then 0 0 the following is true: (1) if (3.7) holds, then 1−2s 1 1 2s 1 − 2s 2s 2s −( R |α | +···+ R |α | ) 1 1 d d |c( F,α)|  e , R = s ; (3.23) r 20 Page 26 of 34 E. Nabizadeh et al. (2) if R ∈ R is given by 1−2s 1 1 2s 1 − 2s 2s 2s −( R |α | +···+ R |α | ) 0,1 1 0,d d |c( F,α)|  e , R = s , (3.24) then (3.7) holds with r in place of r . Proof Let r , r , r ∈ R be such that r < r < r < r . By Lemma 2.7 we may 1 2 3 1 2 3 assume that d = 1, and by Lemma 3.15 the result follows if we prove 1 1 1−2s R ·α 2s 2 −r (logz) 2 sup |c( F,α)e |  | F (z)e | dλ(z) α∈N C 2s R ·α 2 sup |c( F,α)e | , (3.25) α∈N where R , R , R ∈ R satisfy 1 2 3 + 1−2s 2s 1 − 2s R = , j = 1, 2, 3. By Theorem 2.6 we have 1−2s −r (logz) 2 2 | F (z)e | dλ(z)  |c( F,α)ϑ (α)| , α∈N where ∞ 1 1−2s −r (logt ) α ϑ (α) = e t dt . 2α! Let −r (log t ) α θ = > 1 and g (t ) = e t . r ,α 1 − 2s In order to prove (3.25), we need to show that 1 1 2s 2s R α R α 2 1 e  ϑ (α)  e , (3.26) which shall be reached by modifying the proof of (15) in [4]. We have 2 (r −r )(log t ) ϑ (α)  e g (t ) dt r r ,α e Paley–Wiener properties for spaces of entire functions Page 27 of 34 20 (r −r )(log t ) sup(g (t )) e dt  sup(g (t )). r ,α r ,α 1 1 t ≥e e t ≥e By straight-forward computations it follows that g (t ) attains its global maximum r ,α for 1−2s 2s t = exp , (3.27) r ,α θ r and that 1 1 1− 2s − 2 R ·α 2s 1 2s 2s g (t ) = exp r θ (θ − 1)α = e , r ,α r ,α 1 1 and the second inequality in (3.26) follows. In order to prove the first inequality in (3.26), we claim that for some c which is independent of α we have c θ (r −r )(log t ) 2 r ,α 1 −  e . (3.28) r ,α In fact, by (3.27) it follows that −1 lim α t =∞. 2,α α→∞ This together with the fact that r < r give α t 2,α r 2,α c c −c 0 lim 1 − = lim 1 − = (e ) = 1 α→∞ α→∞ t t r r 2,α 2,α and 2s (r −r )((1−2s)α/r ) 2 2 lim e = 0, α→∞ and (3.28) follows. By (3.28) and the fact that > 1 we get 2s r ,α 1 θ 1 θ 2 −r (log t ) α −r (log t ) α r ,α ϑ (α)  e t dt  e (t − c) r r ,α α! α! t −c r ,α 1 1 1 θ 1 2s 2s −r (log t ) α 2 R ·α 2 R ·α 2 r ,α 2 3 e t = e  e . r ,α α! α! This gives the result.   20 Page 28 of 34 E. Nabizadeh et al. 3.3 Proofs of main results Next we prove Proposition 3.9 and thereby Theorem 3.2. Proof of Proposition 3.9 It is clear that (2) ⇒ (3) ⇒ (4) ⇒ (5). By Proposition 3.13 it follows that (5) implies (1). We need to prove that (1) implies (2). Suppose (1) holds and let r = r. Choose r , r , r ∈ R such that r < r < r ≤ 4 1 2 3 0 1 2 2 ∞ d r < r and r r < r , χ ∈ R (C ), and let F be as in Lemma 3.14 with r and 3 4 1 4 α,χ 2 2 r ,r 2 3 r in place of r and r .If τ = 1, then Lemma 3.18 (2) gives 3 1 2 α − |c( F,α)|  r α! . (3.29) Let F (z) = c( F,α)ς z . (3.30) 0 α α∈N We claim that the series in (3.30) is uniformly convergent with respect to z in D (0). In fact, if |z |≤ r , then (3.14)gives j 4, j α d −2α α α d α |c( F,α)ς z |  α r r r =α ρ , 1 4 r r d 1, j 4, j d α where ρ ∈ R satisfies ρ = < 1. Since α ρ is convergent, Weier- j d + α∈N 2, j strass theorem shows that (3.30) is uniformly convergent and defines an analytic function in D (0). Hence, F ∈ A( D (0)). Furthermore, by the support of χ r 0 r 4, j 4, j we get ⎛ ⎞ ⎝ ⎠ ( F · χ) =  c( F,α)ς z · χ A 0 A α α∈N = c( F,α)ς  (z · χ) = c( F,α)e = F . (3.31) α A α d d α∈N α∈N Hence (2) holds, and the proof is complete. For future references we observe that if ς and χ are the same as in Lemma 3.14, then (3.30) shows that the relationship between c( F,α) and c( F ,α) is given by c( F ,α) = c( F,α)ς α! . (3.32) 0 α Proof of Theorem 3.4 The equivalence between (1) and (2) is clear. It is also obvious that (3) implies (2) in view of Proposition 3.9. We shall prove the equivalence between (1) and (3). d d Suppose (1) holds. Then (3.29) holds for every r ∈ R .Let r , R ∈ R be chosen + + ∞ d such that χ ∈ R (C ), F be as in Lemma 3.14 and let F be given by (3.30). By α,χ 0 r , R (3.29)wehave d −2α α |c( F,α)ς |  α r r 1 Paley–Wiener properties for spaces of entire functions Page 29 of 34 20 for every r ∈ R , giving that |c( F,α)ς |  r for every r ∈ R . This implies that the series in (3.30) is locally uniformly convergent d d with respect to z and defines an entire function on C . Hence F ∈ A(C ). Moreover, by (3.31) it follows that  ( F χ) = F, and we have proved that (1) implies (3). A 0 Next suppose that (3) holds. Then |c( F ,α)|  r α! for every r ∈ R .By(3.14) and (3.32) we get 1 1 −1 − α −1 α 2α − 2 2 |c( F,α)|=|c( F ,α)|ς α!  r ς  r R α! . α α Since r ∈ R can be chosen arbitrarily small we get α − |c( F,α)|  r α! d d for every r ∈ R . This implies that F ∈ A (C ). That is, we have proved that (3) 0 0, + 1 implies (1), and the result follows. Next we prove Proposition 3.10. Proof of Proposition 3.10 Suppose (3.3) holds, and let r , r ∈ R be such that r < 1 2 0 r < r < r. Then Lemma 3.18 gives 1 2 2τ +1 ·α 2r −τ |c( F,α)|  α! . 2τ + 1 By Lemma 3.14 and (3.32) we get 2τ +1 ·α 2r −τ d −2α |c( F ,α)|  α! α α!t 2τ + 1 2τ +1   2τ −1 ·α ·α 2 2 2r 2 R 2 2 −(τ −1) −2α −(τ −1) α! t = α! , 2τ + 1 2τ − 1 when 2τ +1 2τ −1 2τ − 1 2r − 2τ −1 R = t . 2 2τ + 1 This proves (1). 20 Page 30 of 34 E. Nabizadeh et al. Suppose instead that r , R , r , r ∈ R are such that (3.6) hold and r < r < 0 0 1 2 0 1 r < r, and that F ∈ A(C ) satisfies (3.4). Also let F =  ( F · χ). Then 2 0 A 0 2τ −1   2τ +1 2 2 2 R 2r 0 1 −2 < t . 2τ − 1 2τ + 1 By combining the latter estimate with Lemma 3.18 we get 2τ +1 ·α 2r −2α 1−τ |c( F ,α)|  t α! 2τ + 1 Hence Lemma 3.14 and (3.32)give 2τ +1 ·α 2r −τ |c( F,α)|  α! 2τ + 1 and Lemma 3.18 again implies that (3.3) holds with r in place of r . This gives the 2 0 result. 1 σ 1 Proof of Theorems 3.5–3.7 First suppose σ ∈ ( , 1), and let σ = and τ = . 2 2σ −1 2σ Then 2σ 2 2σ 2 = and = . σ + 1 2τ + 1 σ − 1 2τ − 1 Theorem 3.5 now follows from these observations and Proposition 3.10 in the case τ ∈ ( , 1). In the same way, Theorem 3.6 follows by choosing τ = 1 in Proposition 3.10. 1 σ 1 Finally, suppose σ ∈ (0, ), and let σ = and τ = . Then 2 1−2σ 2σ 2σ 2 2σ 2 = and = , σ + 1 2τ + 1 σ + 1 2τ − 1 and Theorem 3.7 follows from these observations and Proposition 3.10 in the case τ> 1. Next we prove Propositions 3.11 and 3.12 and thereby Theorem 3.8. Proof of Propositions 3.11 and 3.12 Let r , r , R ∈ R , j = 0, 1, 2, 3, be such that j j 1−2s 2s 1 − 2s r < r < r < r < r and R = s . 0 1 2 3 j First suppose that F ∈ A(C ) satisfies (3.7) and let F be the formal power series expansion with coefficients given by (3.32). Then F =  ( F · χ). A 0 Paley–Wiener properties for spaces of entire functions Page 31 of 34 20 By Lemma 3.19 we get 1 1 2s 2s −( R |α | +···+ R |α | ) 1,1 1 1,d d |c( F,α)|  e . Hence Lemma 3.14 and (3.32)give 1 1 2s 2s −2α −( R |α | +···+ R |α | ) 1,1 1 1,d d |c( F ,α)|  α!t e 1 1 2s 2s −( R |α | +···+ R |α | ) 2,1 1 2,d d e . In the last inequality we have used the fact that s < , which implies that R < R 2 1 and 1 1 2s 2s −2α ( R − R )|α | +···+( R − R )|α | 1,1 2,1 1 1,d 2,d d t α!  e . By applying Lemma 3.19 again it follows that F satisfies (3.8) with r in place of 0 3 r . This gives (1). Suppose instead that F ∈ A(C ) satisfies (3.8) and let F =  ( F · χ). Then 0 A 0 Lemma 3.19 gives 1 1 2s 2s −( R |α | +···+ R |α | ) 1,1 1 1,d d |c( F ,α)|  e , and it follows from Lemma 3.14 and (3.32) that 1 1 2s 2s −1 2α −( R |α | +···+ R |α | ) 1,1 1 1,d d |c( F,α)|  α! t e 1 1 2s 2s −( R |α | +···+ R |α | ) 1,1 1 1,d d e . By applying Lemma 3.19 again we deduce (3.7) with r in place of r , and Proposition 2 0 3.11 follows. Proposition 3.12 is a straight-forward consequence of (3.32). The details are left for the reader. 4 Characterizations of Pilipovic´ spaces In this section we combine Lemma 2.10 with Theorems 3.2–3.8 to get characterizations of Pilipovic´ spaces. We begin with the following characterization of H . The result is a straight-forward combination of Lemma 2.10 and Theorem 3.2. The details are left for the reader. Proposition 4.1 Let φ be as in (2.7),r ∈ R , χ be the characteristic function for D (0) and let s > 0. Then the following conditions are equivalent: (1) f ∈ H (R ); 1 20 Page 32 of 34 E. Nabizadeh et al. ∗  2d (2) f = V Ffor some F ∈ E (R ); φ s ∗  2d ∞ 2d (3) f = V Ffor some F ∈ E (R ) ∩ L (R ); ∗  2d ∞ 2d (4) f = V Ffor some F ∈ E (R ) ∩ L (R ) which satisfies −i x ,ξ F (x,ξ) = F (x − iξ)e χ(x,ξ) (4.1) for some r ∈ R , χ = χ and F ∈ A( D (0)). r 0 r ∞ d Remark 4.2 It is clear that χ in Proposition 4.1 can be chosen as any χ ∈ R (C ) with suitable support properties. 1 2 2 (|x | +|ξ | ) Remark 4.3 In (2.32) there is a factor e which is absent in (4.1). We notice ∞ d that this factor is not needed in (4.1) because R (C ) is invariant under multiplica- t ,t 1 2 tions of such functions. The next results follow from Lemma 2.10 and Theorems 3.4–3.8. The details are left for the reader. ∞ d Proposition 4.4 Let φ be as in (2.7) and χ ∈ R (C ). Then the following conditions are equivalent: (1) f ∈ H (R ); 0, ∗  2d ∞ 2d (2) f = V Ffor some F ∈ E (R ) ∩ L (R ) which satisfies (4.1) for some F ∈ A(C ). ∞ d 1 Proposition 4.5 Let φ be as in (2.7), χ ∈ R (C ), σ ∈ ( , 1) and let σ = . 2σ − 1 Then the following conditions are equivalent: d d (1) f ∈ H (R )( f ∈ H (R )); 0, σ σ ∗  2d ∞ 2d (2) f = V Ffor some F ∈ E (R ) ∩ L (R ) which satisfies (4.1) for some d  d F ∈ A (C )( F ∈ A (C )). 0 0 0, σ σ 0 0 ∞ d 1 Proposition 4.6 Let φ be as in (2.7), χ ∈ R (C ) and let σ = . Then the following conditions are equivalent: d d (1) f ∈ H (R )( f ∈ H (R )); 0, σ σ 2d ∞ 2d (2) f = V Ffor some F ∈ E (R ) ∩ L (R ) which satisfies (4.1) for some d d F ∈ A (C )( F ∈ A (C )). 0 0 0,1/2 0,1/2 Paley–Wiener properties for spaces of entire functions Page 33 of 34 20 ∞ d Proposition 4.7 Let φ be as in (2.7), χ ∈ R (C ), σ ∈ (0, ) and let σ = . 1 − 2σ Then the following conditions are equivalent: d d (1) f ∈ H (R )( f ∈ H (R )); 0, σ σ ∗  2d ∞ 2d (2) f = V Ffor some F ∈ E (R ) ∩ L (R ) which satisfies (4.1) for some d d F ∈ A (C )( F ∈ A (C )). 0  0 0, σ σ 0 0 ∞ d 1 Proposition 4.8 Let φ be as in (2.7), χ ∈ R (C ) and let s ∈ (0, ). Then the following conditions are equivalent: d d (1) f ∈ H (R )( f ∈ H (R )); s s ∗  2d ∞ 2d (2) f = V Ffor some F ∈ E (R ) ∩ L (R ) which satisfies (4.1) for some d d F ∈ A (C )( F ∈ A (C )). 0 s 0 0,s Acknowledgements Open access funding provided by Linnaeus University. Compliance with ethical standards Conflict of interest There are no conflict of interests. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. References 1. 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