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Hilbert space valued Gabor frames in weighted amalgam spaces

Authors
  • Poria, Anirudha
  • Swain, Jitendriya
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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