# Hilbert space valued Gabor frames in weighted amalgam spaces

Hilbert space valued Gabor frames in weighted amalgam spaces AbstractLet ℍ{\mathbb{H}} be a separable Hilbert space. In this paper, we establish a generalization of Walnut’s representation and Janssen’s representation of the ℍ{\mathbb{H}}-valued Gabor frame operator on ℍ{\mathbb{H}}-valued weighted amalgam spaces Wℍ⁢(Lp,Lvq){W_{\mathbb{H}}(L^{p},L^{q}_{v})}, 1≤p,q≤∞{1\leq p,q\leq\infty}. Also, we show that the frame operator is invertible on Wℍ⁢(Lp,Lvq){W_{\mathbb{H}}(L^{p},L^{q}_{v})}, 1≤p,q≤∞{1\leq p,q\leq\infty}, if the window function is in the Wiener amalgam space Wℍ⁢(L∞,Lw1){W_{\mathbb{H}}(L^{\infty},L^{1}_{w})}. Further, we obtain the Walnut representation and invertibility of the frame operator corresponding to Gabor superframes and multi-window Gabor frames on Wℍ⁢(Lp,Lvq){W_{\mathbb{H}}(L^{p},L^{q}_{v})}, 1≤p,q≤∞{1\leq p,q\leq\infty}, as a special case by choosing the appropriate Hilbert space ℍ{\mathbb{H}}. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Pure and Applied Mathematics de Gruyter

# Hilbert space valued Gabor frames in weighted amalgam spaces

, Volume 10 (4): 18 – Oct 1, 2019
18 pages

/lp/de-gruyter/hilbert-space-valued-gabor-frames-in-weighted-amalgam-spaces-KJQ0YnW0hg
Publisher
de Gruyter
© 2019 Walter de Gruyter GmbH, Berlin/Boston
ISSN
1869-6090
eISSN
1869-6090
DOI
10.1515/apam-2018-0067
Publisher site
See Article on Publisher Site

### Abstract

AbstractLet ℍ{\mathbb{H}} be a separable Hilbert space. In this paper, we establish a generalization of Walnut’s representation and Janssen’s representation of the ℍ{\mathbb{H}}-valued Gabor frame operator on ℍ{\mathbb{H}}-valued weighted amalgam spaces Wℍ⁢(Lp,Lvq){W_{\mathbb{H}}(L^{p},L^{q}_{v})}, 1≤p,q≤∞{1\leq p,q\leq\infty}. Also, we show that the frame operator is invertible on Wℍ⁢(Lp,Lvq){W_{\mathbb{H}}(L^{p},L^{q}_{v})}, 1≤p,q≤∞{1\leq p,q\leq\infty}, if the window function is in the Wiener amalgam space Wℍ⁢(L∞,Lw1){W_{\mathbb{H}}(L^{\infty},L^{1}_{w})}. Further, we obtain the Walnut representation and invertibility of the frame operator corresponding to Gabor superframes and multi-window Gabor frames on Wℍ⁢(Lp,Lvq){W_{\mathbb{H}}(L^{p},L^{q}_{v})}, 1≤p,q≤∞{1\leq p,q\leq\infty}, as a special case by choosing the appropriate Hilbert space ℍ{\mathbb{H}}.

### Journal

Advances in Pure and Applied Mathematicsde Gruyter

Published: Oct 1, 2019