On the frame set of the second-order B-spline

A. Ganiou D. Atindehou; Christina Frederick; Yébéni B. Kouagou; Kasso A. Okoudjou

doi:10.1016/j.acha.2022.08.007

On the frame set of the second-order B-spline

A. Ganiou D. Atindehou, Christina Frederick, Yébéni B. Kouagou, Kasso A. Okoudjou

Mathematical Sciences

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1 Scopus citations

Abstract

The frame set of a function g∈L²(R) is the set of all parameters (a,b)∈R₊² for which the collection of time-frequency shifts of g along aZ×bZ form a Gabor frame for L²(R). Finding the frame set of a given function remains a challenging open problem in time-frequency analysis. In this paper, we establish new regions of the frame set of the second-order B-spline. Our method uses the compact support of this function to partition a subset of the putative frame set and finds an explicit dual window function in each subregion. Numerical evidence indicates the existence of further regions belonging to the frame set.

Original language	English (US)
Pages (from-to)	237-250
Number of pages	14
Journal	Applied and Computational Harmonic Analysis
Volume	62
DOIs	https://doi.org/10.1016/j.acha.2022.08.007
State	Published - Jan 2023

All Science Journal Classification (ASJC) codes

Applied Mathematics

Keywords

B-splines
Frame set
Gabor frames

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10.1016/j.acha.2022.08.007

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title = "On the frame set of the second-order B-spline",

abstract = "The frame set of a function g∈L2(R) is the set of all parameters (a,b)∈R+2 for which the collection of time-frequency shifts of g along aZ×bZ form a Gabor frame for L2(R). Finding the frame set of a given function remains a challenging open problem in time-frequency analysis. In this paper, we establish new regions of the frame set of the second-order B-spline. Our method uses the compact support of this function to partition a subset of the putative frame set and finds an explicit dual window function in each subregion. Numerical evidence indicates the existence of further regions belonging to the frame set.",

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T1 - On the frame set of the second-order B-spline

AU - Atindehou, A. Ganiou D.

AU - Frederick, Christina

AU - Kouagou, Yébéni B.

AU - Okoudjou, Kasso A.

PY - 2023/1

Y1 - 2023/1

N2 - The frame set of a function g∈L2(R) is the set of all parameters (a,b)∈R+2 for which the collection of time-frequency shifts of g along aZ×bZ form a Gabor frame for L2(R). Finding the frame set of a given function remains a challenging open problem in time-frequency analysis. In this paper, we establish new regions of the frame set of the second-order B-spline. Our method uses the compact support of this function to partition a subset of the putative frame set and finds an explicit dual window function in each subregion. Numerical evidence indicates the existence of further regions belonging to the frame set.

AB - The frame set of a function g∈L2(R) is the set of all parameters (a,b)∈R+2 for which the collection of time-frequency shifts of g along aZ×bZ form a Gabor frame for L2(R). Finding the frame set of a given function remains a challenging open problem in time-frequency analysis. In this paper, we establish new regions of the frame set of the second-order B-spline. Our method uses the compact support of this function to partition a subset of the putative frame set and finds an explicit dual window function in each subregion. Numerical evidence indicates the existence of further regions belonging to the frame set.

KW - B-splines

KW - Frame set

KW - Gabor frames

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SP - 237

EP - 250

JO - Applied and Computational Harmonic Analysis

JF - Applied and Computational Harmonic Analysis

ER -

On the frame set of the second-order B-spline

Abstract

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