关于由 B-样条函数、全正函数和赫米特函数生成的 Gabor 框架

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics Pub Date : 2024-08-30 DOI:10.1016/j.apnum.2024.08.021

Riya Ghosh, A. Antony Selvan

引用次数: 0

摘要

窗口ϕ∈L2(R)的框架集是所有网格参数（α,β）∈R+2 的子集，使得 G(j,α,β)={e2πiβm⋅j(⋅-αk):k,m∈Z} 构成 L2(R) 的框架。本文研究了 B-样条函数、全正函数和 Hermite 函数的框架集。我们利用移位不变空间中的采样理论与 Gabor 分析之间的联系，推导出 Gabor 框架的充分条件。因此，我们得到了属于 B-样条函数和 Hermite 函数框架集的新框架区域。对于包括某些全正函数在内的一类函数，我们证明，对于任意选择的网格参数 α,β>0 与 αβ<1 ，存在一个取决于 αβ 的 γ>0 ，这样 G(ϕ(γ⋅),α,β) 就形成了 L2(R) 的框架。我们的结果给出了明确的框架边界。

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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On Gabor frames generated by B-splines, totally positive functions, and Hermite functions

The frame set of a window $ϕ \in L^{2} (R)$ is the subset of all lattice parameters $(α, β) \in R_{+}^{2}$ such that $G (ϕ, α, β) = {e^{2 π i β m \cdot} ϕ (\cdot - α k) : k, m \in Z}$ forms a frame for $L^{2} (R)$ . In this paper, we investigate the frame set of B-splines, totally positive functions, and Hermite functions. We derive a sufficient condition for Gabor frames using the connection between sampling theory in shift-invariant spaces and Gabor analysis. As a consequence, we obtain a new frame region belonging to the frame set of B-splines and Hermite functions. For a class of functions that includes certain totally positive functions, we prove that for any choice of lattice parameters $α, β > 0$ with $α β < 1$ , there exists a $γ > 0$ depending on αβ such that $G (ϕ (γ \cdot), α, β)$ forms a frame for $L^{2} (R)$ . Our results give explicit frame bounds.

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来源期刊

Applied Numerical Mathematics 数学-应用数学

CiteScore

5.60

自引率

7.10%

发文量

225

审稿时长

7.2 months

期刊介绍： The purpose of the journal is to provide a forum for the publication of high quality research and tutorial papers in computational mathematics. In addition to the traditional issues and problems in numerical analysis, the journal also publishes papers describing relevant applications in such fields as physics, fluid dynamics, engineering and other branches of applied science with a computational mathematics component. The journal strives to be flexible in the type of papers it publishes and their format. Equally desirable are: (i) Full papers, which should be complete and relatively self-contained original contributions with an introduction that can be understood by the broad computational mathematics community. Both rigorous and heuristic styles are acceptable. Of particular interest are papers about new areas of research, in which other than strictly mathematical arguments may be important in establishing a basis for further developments. (ii) Tutorial review papers, covering some of the important issues in Numerical Mathematics, Scientific Computing and their Applications. The journal will occasionally publish contributions which are larger than the usual format for regular papers. (iii) Short notes, which present specific new results and techniques in a brief communication.