Linear Independence of Finite Gabor Systems Determined by Behavior at Infinity

Johnny J. Benedetto; Abdelkrim Bourouihiya

doi:10.1007/s12220-013-9423-8

Linear Independence of Finite Gabor Systems Determined by Behavior at Infinity

Johnny J. Benedetto
, Abdelkrim Bourouihiya

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

8 Scopus citations

Abstract

We prove that the HRT (Heil, Ramanathan, and Topiwala) conjecture holds for finite Gabor systems generated by square-integrable functions with certain behavior at infinity. These functions include functions ultimately decaying faster than any exponential function, as well as square-integrable functions ultimately analytic and whose germs are in a Hardy field that is closed under translations. Two classes of the latter type of functions are the set of square-integrable logarithmico-exponential functions and the set of square-integrable Pfaffian functions. We also prove the HRT conjecture for certain finite Gabor systems generated by positive functions.

Original language	American English
Pages (from-to)	226-254
Number of pages	29
Journal	Journal of Geometric Analysis
Volume	25
Issue number	1
DOIs	https://doi.org/10.1007/s12220-013-9423-8
State	Published - Jan 1 2015

Bibliographical note

Publisher Copyright:
© 2013, Mathematica Josephina, Inc.

Funding

The authors gratefully acknowledge the support from the AFOSR-MURI Grant FA9550-05-1-0443. The first-named author is also appreciative of the support of ARO-MURI Grant W911NF-09-1-0383 and NGA Grant HM-1582-08-1-0009. The authors also had the good fortune to obtain expert background and insights from Chris Heil. We note that the results herein were all obtained by early 2008. The authors were motivated to publish upon hearing a beautiful lecture in 2012 by Darrin Speegle reporting on his work [] with Marcin Bownik on the HRT conjecture. Further, we received invaluable assistance from Travis Andrews. Finally, we want to thank the referee for an authoritative, constructive, incisive, thorough review.

Funders	Funder number
AFOSR-MURI	FA9550-05-1-0443
ARO-MURI	W911NF-09-1-0383
Air Force Office of Scientific Research
Army Research Office
National Geospatial-Intelligence Agency
National Geospatial-Intelligence Agency	HM-1582-08-1-0009
National Geospatial-Intelligence Agency

ASJC Scopus Subject Areas

Geometry and Topology

Keywords

Gabor systems
HRT conjecture
Hardy fields
Kronecker's theorem
Kronecker’s theorem

Disciplines

Mathematics

Access to Document

10.1007/s12220-013-9423-8

Cite this

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abstract = " We prove that the HRT (Heil, Ramanathan, and Topiwala) conjecture holds for finite Gabor systems generated by square-integrable functions with certain behavior at infinity. These functions include functions ultimately decaying faster than any exponential function, as well as square-integrable functions ultimately analytic and whose germs are in a Hardy field that is closed under translations. Two classes of the latter type of functions are the set of square-integrable logarithmico-exponential functions and the set of square-integrable Pfaffian functions. We also prove the HRT conjecture for certain finite Gabor systems generated by positive functions.",

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N2 - We prove that the HRT (Heil, Ramanathan, and Topiwala) conjecture holds for finite Gabor systems generated by square-integrable functions with certain behavior at infinity. These functions include functions ultimately decaying faster than any exponential function, as well as square-integrable functions ultimately analytic and whose germs are in a Hardy field that is closed under translations. Two classes of the latter type of functions are the set of square-integrable logarithmico-exponential functions and the set of square-integrable Pfaffian functions. We also prove the HRT conjecture for certain finite Gabor systems generated by positive functions.

AB - We prove that the HRT (Heil, Ramanathan, and Topiwala) conjecture holds for finite Gabor systems generated by square-integrable functions with certain behavior at infinity. These functions include functions ultimately decaying faster than any exponential function, as well as square-integrable functions ultimately analytic and whose germs are in a Hardy field that is closed under translations. Two classes of the latter type of functions are the set of square-integrable logarithmico-exponential functions and the set of square-integrable Pfaffian functions. We also prove the HRT conjecture for certain finite Gabor systems generated by positive functions.

KW - Gabor systems

KW - HRT conjecture

KW - Hardy fields

KW - Kronecker's theorem

KW - Kronecker’s theorem

UR - https://nsuworks.nova.edu/math_facarticles/70

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JO - Journal of Geometric Analysis

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IS - 1

ER -

Linear Independence of Finite Gabor Systems Determined by Behavior at Infinity

Abstract

Bibliographical note

Funding

ASJC Scopus Subject Areas

Keywords

Disciplines

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