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

TL;DR

This paper investigates nonstationary frames of translates, Gabor frames, and wavelet frames for specific groups, providing conditions for their existence, structure, and properties like Riesz basis and linear independence.

Contribution

It establishes necessary and sufficient conditions for nonstationary frames of translates and explores their structure and properties for the Weyl--Heisenberg and extended affine groups.

Findings

Conditions for existence of nonstationary frames of translates

Representation of functions via Fourier transform of window functions

Characterization of Riesz bases among nonstationary frames

Abstract

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.