Nonstationary frames of translates and frames for the Weyl--Heisenberg group and the extended affine group

In this work, we analyze Gabor frames for the Weyl--Heisenberg group and wavelet frames for the extended affine group. Firstly, we give necessary and sufficient conditions for the existence of nonstationary frames of translates. Using these conditions, we give the existence of Gabor frames from the...

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Bibliographic Details
Published in:arXiv.org 2023-06
Main Authors: Jindal, Divya, Vashisht, Lalit Kumar
Format: Article
Language:English
Subjects:
Affine transformations
Algorithms
Approximation
Fourier transforms
Frames
Mathematical analysis
Window functions
Online Access:Get full text
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Summary:In this work, we analyze Gabor frames for the Weyl--Heisenberg group and wavelet frames for the extended affine group. Firstly, we give necessary and sufficient conditions for the existence of nonstationary frames of translates. Using these conditions, we give the existence of Gabor frames from the Weyl--Heisenberg group and wavelet frames for the extended affine group. We present a representation of functions in the closure of the linear span of a Gabor frame sequence in terms of the Fourier transform of window functions. We show that the canonical dual of frames of translates has the same structure. An approximation of inverse of the frame operator of nonstationary frames of translates is presented. It is shown that a nonstationary frame of translates is a Riesz basis if it is linearly independent and satisfies approximation of the inverse frame operator. Finally, we give equivalent conditions for a nonstationary sequence of translates to be linearly independent.
ISSN:2331-8422
DOI:10.48550/arxiv.2209.15054