# Hilbert space valued Gabor frames in weighted amalgam spaces

Authors
Type
Published Article
Journal
Advances in Pure and Applied Mathematics
Publisher
De Gruyter
Publication Date
Aug 23, 2018
Volume
10
Issue
4
Pages
377–394
Identifiers
DOI: 10.1515/apam-2018-0067
Source
De Gruyter
Keywords
License
Yellow

## Abstract

Let 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 W ℍ ⁢ ( L p , L v q ) {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 W ℍ ⁢ ( L p , L v q ) {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 W ℍ ⁢ ( L ∞ , L w 1 ) {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 W ℍ ⁢ ( L p , L v q ) {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}} .

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