# A Hilbert C\*-module for Gabor systems

Source: [A Hilbert C\*-module for Gabor systems](<https://www.sota2.com/research/paper/68fa4c941e209131ce94d6b7>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-25T08:50:14.678Z

Authors: Michael Coco, M.C. Lammers

Published: 2001-02-20T20:08:51.000Z

## Abstract

We construct Hilbert $C^\*$-modules useful for studying Gabor systems and show that they are Banach algebras under pointwise multiplication. For rational $ab\<1$ we prove that the set of functions $g \\in L^2(R)$ so that $(g,a,b)$ is a Bessel system is an ideal for the Hilbert $C^\*$-module given this pointwise algebraic structure. This allows us to give a multiplicative perturbation theorem for frames. Finally we show that a system $(g,a,b)$ yields a frame for $L^2(R)$ iff it is a modular frame for the given Hilbert $C^\*$-module.

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/math/0102165>)

[PDF](<https://arxiv.org/pdf/math/0102165.pdf>)