# Gabor Frames in ℓ2(Z) and Linear Dependence

Ole Christensen*, Marzieh Hasannasab

*Corresponding author for this work

Research output: Contribution to journalJournal articleResearchpeer-review

### Abstract

We prove that an overcomplete Gabor frame in (Formula presented.) generated by a finitely supported sequence is always linearly dependent. This is a particular case of a general result about linear dependence versus independence for Gabor systems in (Formula presented.) with modulation parameter 1 / M and translation parameter N for some (Formula presented.) and generated by a finite sequence g in (Formula presented.) with K nonzero entries.
Original language English Journal of Fourier Analysis and Applications 25 1 101–107 1069-5869 https://doi.org/10.1007/s00041-017-9572-4 Published - 2018

### Keywords

• Frames
• Gabor system in $${\ell }^2({\mathbb {Z}})$$â2(Z)
• Linear dependency of Gabor systems

### Cite this

Christensen, Ole ; Hasannasab, Marzieh. / Gabor Frames in ℓ2(Z) and Linear Dependence. In: Journal of Fourier Analysis and Applications. 2018 ; Vol. 25, No. 1. pp. 101–107.
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abstract = "We prove that an overcomplete Gabor frame in (Formula presented.) generated by a finitely supported sequence is always linearly dependent. This is a particular case of a general result about linear dependence versus independence for Gabor systems in (Formula presented.) with modulation parameter 1 / M and translation parameter N for some (Formula presented.) and generated by a finite sequence g in (Formula presented.) with K nonzero entries.",
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Gabor Frames in ℓ2(Z) and Linear Dependence. / Christensen, Ole; Hasannasab, Marzieh.

In: Journal of Fourier Analysis and Applications, Vol. 25, No. 1, 2018, p. 101–107.

Research output: Contribution to journalJournal articleResearchpeer-review

TY - JOUR

T1 - Gabor Frames in ℓ2(Z) and Linear Dependence

AU - Christensen, Ole

AU - Hasannasab, Marzieh

PY - 2018

Y1 - 2018

N2 - We prove that an overcomplete Gabor frame in (Formula presented.) generated by a finitely supported sequence is always linearly dependent. This is a particular case of a general result about linear dependence versus independence for Gabor systems in (Formula presented.) with modulation parameter 1 / M and translation parameter N for some (Formula presented.) and generated by a finite sequence g in (Formula presented.) with K nonzero entries.

AB - We prove that an overcomplete Gabor frame in (Formula presented.) generated by a finitely supported sequence is always linearly dependent. This is a particular case of a general result about linear dependence versus independence for Gabor systems in (Formula presented.) with modulation parameter 1 / M and translation parameter N for some (Formula presented.) and generated by a finite sequence g in (Formula presented.) with K nonzero entries.

KW - Frames

KW - Gabor system in $${\ell }^2({\mathbb {Z}})$$â2(Z)

KW - Linear dependency of Gabor systems

U2 - 10.1007/s00041-017-9572-4

DO - 10.1007/s00041-017-9572-4

M3 - Journal article

VL - 25

SP - 101

EP - 107

JO - Journal of Fourier Analysis and Applications

JF - Journal of Fourier Analysis and Applications

SN - 1069-5869

IS - 1

ER -