On the non-frame property of Gabor systems with Hermite generators and the frame set conjecture

IF 2.6 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis Pub Date : 2025-01-10 DOI:10.1016/j.acha.2025.101747

Andreas Horst , Jakob Lemvig , Allan Erlang Videbæk

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引用次数: 0

Abstract

The frame set conjecture for Hermite functions formulated in [13] states that the Gabor frame set for these generators is the largest possible, that is, the time-frequency shifts of the Hermite functions associated with sampling rates α and modulation rates β that avoid all known obstructions lead to Gabor frames for

L^{2} (R)

. By results in [24], [25] and [22], it is known that the conjecture is true for the Gaussian, the 0th order Hermite functions, and false for Hermite functions of order

2, 3, 6, 7, 10, 11, \dots

, respectively. In this paper we disprove the remaining cases except for the 1st order Hermite function.

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带有Hermite发生器的Gabor系统的非框架性质及框架集猜想

在[13]中表述的Hermite函数的帧集猜想表明，这些发生器的Gabor帧集是可能最大的，也就是说，与采样率α和调制率β相关的Hermite函数的时频移可以避免所有已知的障碍，从而导致L2(R)的Gabor帧。由[24,25]和[22]的结果可知，该猜想对高斯、0阶Hermite函数为真，对2、3、6、7、10、11、…阶Hermite函数为假。本文证明了除一阶Hermite函数外的其他情况。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Applied and Computational Harmonic Analysis 物理-物理：数学物理

CiteScore

5.40

自引率

4.00%

发文量

审稿时长

22.9 weeks

期刊介绍： Applied and Computational Harmonic Analysis (ACHA) is an interdisciplinary journal that publishes high-quality papers in all areas of mathematical sciences related to the applied and computational aspects of harmonic analysis, with special emphasis on innovative theoretical development, methods, and algorithms, for information processing, manipulation, understanding, and so forth. The objectives of the journal are to chronicle the important publications in the rapidly growing field of data representation and analysis, to stimulate research in relevant interdisciplinary areas, and to provide a common link among mathematical, physical, and life scientists, as well as engineers.