带 Haar 窗口的 Gabor 系统框架集

IF 2.6 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis Pub Date : 2024-03-18 DOI:10.1016/j.acha.2024.101655

Xin-Rong Dai , Meng Zhu

{"title":"带 Haar 窗口的 Gabor 系统框架集","authors":"Xin-Rong Dai , Meng Zhu","doi":"10.1016/j.acha.2024.101655","DOIUrl":null,"url":null,"abstract":"<div>We describe the full structure of the frame set for the Gabor system <math><mi>G</mi><mo>(</mo><mi>g</mi><mo>;</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>)</mo><mo>:</mo><mo>=</mo><mo>{</mo><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo><mn>2</mn><mi>π</mi><mi>i</mi><mi>m</mi><mi>β</mi><mo>⋅</mo></mrow></msup><mi>g</mi><mo>(</mo><mo>⋅</mo><mo>−</mo><mi>n</mi><mi>α</mi><mo>)</mo><mo>:</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>∈</mo><mi>Z</mi><mo>}</mo></math> with the window being the Haar function <math><mi>g</mi><mo>=</mo><mo>−</mo><msub><mrow><mi>χ</mi></mrow><mrow><mo>[</mo><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>,</mo><mn>0</mn><mo>)</mo></mrow></msub><mo>+</mo><msub><mrow><mi>χ</mi></mrow><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>)</mo></mrow></msub></math>. This is the first compactly supported window function for which the frame set is represented explicitly.The strategy of this paper is to introduce the piecewise linear transformation <math><mi>M</mi></math> on the unit circle, and to provide a complete characterization of structures for its (symmetric) maximal invariant sets. This transformation is related to the famous three gap theorem of Steinhaus which may be of independent interest. Furthermore, a classical criterion on Gabor frames is improved, which allows us to establish a necessary and sufficient condition for the Gabor system <math><mi>G</mi><mo>(</mo><mi>g</mi><mo>;</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>)</mo></math> to be a frame, i.e., the symmetric invariant set of the transformation <math><mi>M</mi></math> is empty.</div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"71 ","pages":"Article 101655"},"PeriodicalIF":2.6000,"publicationDate":"2024-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Frame set for Gabor systems with Haar window\",\"authors\":\"Xin-Rong Dai , Meng Zhu\",\"doi\":\"10.1016/j.acha.2024.101655\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div>We describe the full structure of the frame set for the Gabor system <math><mi>G</mi><mo>(</mo><mi>g</mi><mo>;</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>)</mo><mo>:</mo><mo>=</mo><mo>{</mo><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo><mn>2</mn><mi>π</mi><mi>i</mi><mi>m</mi><mi>β</mi><mo>⋅</mo></mrow></msup><mi>g</mi><mo>(</mo><mo>⋅</mo><mo>−</mo><mi>n</mi><mi>α</mi><mo>)</mo><mo>:</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>∈</mo><mi>Z</mi><mo>}</mo></math> with the window being the Haar function <math><mi>g</mi><mo>=</mo><mo>−</mo><msub><mrow><mi>χ</mi></mrow><mrow><mo>[</mo><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>,</mo><mn>0</mn><mo>)</mo></mrow></msub><mo>+</mo><msub><mrow><mi>χ</mi></mrow><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>)</mo></mrow></msub></math>. This is the first compactly supported window function for which the frame set is represented explicitly.The strategy of this paper is to introduce the piecewise linear transformation <math><mi>M</mi></math> on the unit circle, and to provide a complete characterization of structures for its (symmetric) maximal invariant sets. This transformation is related to the famous three gap theorem of Steinhaus which may be of independent interest. Furthermore, a classical criterion on Gabor frames is improved, which allows us to establish a necessary and sufficient condition for the Gabor system <math><mi>G</mi><mo>(</mo><mi>g</mi><mo>;</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>)</mo></math> to be a frame, i.e., the symmetric invariant set of the transformation <math><mi>M</mi></math> is empty.</div>\",\"PeriodicalId\":55504,\"journal\":{\"name\":\"Applied and Computational Harmonic Analysis\",\"volume\":\"71 \",\"pages\":\"Article 101655\"},\"PeriodicalIF\":2.6000,\"publicationDate\":\"2024-03-18\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Applied and Computational Harmonic Analysis\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S1063520324000320\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied and Computational Harmonic Analysis","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1063520324000320","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}

引用次数: 0

摘要

我们描述了以 Haar 函数为窗口的 Gabor 系统帧集的完整结构。这是第一个明确表示帧集的紧凑支持窗口函数。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

Frame set for Gabor systems with Haar window

We describe the full structure of the frame set for the Gabor system $G (g; α, β) : = {e^{- 2 π i m β \cdot} g (\cdot - n α) : m, n \in Z}$ with the window being the Haar function $g = - χ_{[- 1 / 2, 0)} + χ_{[0, 1 / 2)}$ . This is the first compactly supported window function for which the frame set is represented explicitly.

The strategy of this paper is to introduce the piecewise linear transformation $M$ on the unit circle, and to provide a complete characterization of structures for its (symmetric) maximal invariant sets. This transformation is related to the famous three gap theorem of Steinhaus which may be of independent interest. Furthermore, a classical criterion on Gabor frames is improved, which allows us to establish a necessary and sufficient condition for the Gabor system $G (g; α, β)$ to be a frame, i.e., the symmetric invariant set of the transformation $M$ is empty.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

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.