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X. Dai, Y. Diao (2006)
The Path-Connectivity of s-Elementary Tight Frame Wavelets
S. Bildea, D. Dutkay, Gabriel Picioroaga (2004)
MRA super-waveletsarXiv: Functional Analysis
X. Dai, Y. Diao, Qing Gu, D. Han (2002)
Frame wavelets in subspaces of ²(ℝ^{}), 130
D. Han, D. Larson (2000)
Frames, bases, and group representations
K. Gröchenig, Yurii Lyubarskii (2008)
Gabor (super)frames with Hermite functionsMathematische Annalen, 345
(2009)
Lyubarskii, Gabor (super) frames with Hermite Functions, Math. Ann
M. Paluszynski, H. Šikić, G. Weiss, Shaoliang Xiao (2003)
Tight Frame Wavelets, their Dimension Functions, MRA Tight Frame Wavelets and Connectivity PropertiesAdvances in Computational Mathematics, 18
X. Dai, Y. Diao, Qing Gu (2001)
Subspaces with normalized tight frame wavelets in R, 130
X. Dai, Y. Diao, Zhongyan Li (2009)
The Path-Connectivity of s-Elementary Frame Wavelets with Frame MRAActa Applicandae Mathematicae, 107
Z. Li, X. Dai, Y. Diao, J. Xin (2010)
Multipliers, phases and connectivity of wavelets in L 2(ℝ2)J. Fourier Anal. Appl., 16
Zhongyan Li, X. Dai, Y. Diao, Wei Huang (2005)
The Path-Connectivity of MRA Wavelets in L 2 (R d )Illinois Journal of Mathematics, 54
X. Dai, Y. Diao (2002)
Frame wavelets in subspaces of L^2(R^d), 130
(2000)
Basis , Frames , and Group representations
D. Dutkay, P. Jorgensen (2005)
Oversampling generates super-wavelets, 135
Qing Gu, D. Han (2005)
Super-Wavelets and Decomposable Wavelet FramesJournal of Fourier Analysis and Applications, 11
D. Speegle (1998)
THE S-ELEMENTARY WAVELETS ARE PATH-CONNECTED
X. Dai
The s-elementary Frame Wavelets are Path Connected
X. Dai, D. Larson (1998)
Wandering Vectors for Unitary Systems and Orthogonal Wavelets
W. Consortium (1998)
Basic properties of waveletsJ. Fourier Anal. Appl., 4
(1998)
Wavelets, their phases, Multipliers and Connectivity
Zhongyan Li, X. Dai, Y. Diao (2010)
Intrinsic s-elementary Parseval frame multiwavelets in L2(Rd)Journal of Mathematical Analysis and Applications, 367
Zhongyan Li, X. Dai, Y. Diao, J. Xin (2010)
Multipliers, Phases and Connectivity of MRA Wavelets in L2(ℝ2)Journal of Fourier Analysis and Applications, 16
X. Dai, Y. Diao, Qing Gu (2000)
Frame wavelet sets in ℝ, 129
H. Sakai (1994)
Basic Properties of WaveletsMedical imaging technology, 12
A super wavelet of length n is an n-tuple (ψ 1,ψ 2,…,ψ n ) in the product space $\prod_{j=1}^{n} L^{2}(\mathbb{R})$ , such that the coordinated dilates of all its coordinated translates form an orthonormal basis for $\prod_{j=1}^{n} L^{2} (\mathbb{R})$ . This concept is generalized to the so-called super frame wavelets, super tight frame wavelets and super normalized tight frame wavelets (or super Parseval frame wavelets), namely an n-tuple (η 1,η 2,…,η n ) in $\prod_{j=1}^{n}L^{2} (\mathbb{R})$ such that the coordinated dilates of all its coordinated translates form a frame, a tight frame, or a normalized tight frame for $\prod_{j=1}^{n} L^{2}(\mathbb{R})$ . In this paper, we study the super frame wavelets and the super tight frame wavelets whose Fourier transforms are defined by set theoretical functions (called s-elementary frame wavelets). An m-tuple of sets (E 1,E 2,…,E m ) is said to be τ-disjoint if the E j ’s are pair-wise disjoint under the 2π-translations. We prove that a τ-disjoint m-tuple (E 1,E 2,…,E m ) of frame sets (i.e., η j defined by $\widehat{\eta_{j}}=\frac{1}{\sqrt{2\pi}}\chi_{E_{j}}$ is a frame wavelet for L 2(ℝ) for each j) lead to a super frame wavelet (η 1,η 2,…,η m ) for $\prod_{j=1}^{m} L^{2} (\mathbb{R})$ where $\widehat{\eta_{j}}=\frac{1}{\sqrt{2\pi}}\chi_{E_{j}}$ . In the case of super tight frame wavelets, we prove that (η 1,η 2,…,η m ), defined by $\widehat{\eta_{j}}=\frac{1}{\sqrt{2\pi}}\chi_{E_{j}}$ , is a super tight frame wavelet for ∏1≤j≤m L 2(ℝ) with frame bound k 0 if and only if each η j is a tight frame wavelet for L 2(ℝ) with frame bound k 0 and that (E 1,E 2,…,E m ) is τ-disjoint. Denote the set of all τ-disjoint s-elementary super frame wavelets for ∏1≤j≤m L 2(ℝ) by $\mathfrak{S}(m)$ and the set of all s-elementary super tight frame wavelets (with the same frame bound k 0) for ∏1≤j≤m L 2(ℝ) by $\mathfrak{S}^{k_{0}}(m)$ . We further prove that $\mathfrak{S}(m)$ and $\mathfrak{S}^{k_{0}}(m)$ are both path-connected under the ∏1≤j≤m L 2(ℝ) norm, for any given positive integers m and k 0.
Acta Applicandae Mathematicae – Springer Journals
Published: Jul 26, 2011
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