{"title":"傅里叶变换下的特征测度","authors":"Michael Baake, Timo Spindeler, Nicolae Strungaru","doi":"10.1007/s00041-023-10045-z","DOIUrl":null,"url":null,"abstract":"Abstract Several classes of tempered measures are characterised that are eigenmeasures of the Fourier transform, the latter viewed as a linear operator on (generally unbounded) Radon measures on $$\\mathbb {R}\\hspace{0.5pt}^d$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>R</mml:mi> <mml:msup> <mml:mspace /> <mml:mi>d</mml:mi> </mml:msup> </mml:mrow> </mml:math> . In particular, we classify all periodic eigenmeasures on $$\\mathbb {R}\\hspace{0.5pt}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>R</mml:mi> <mml:mspace /> </mml:mrow> </mml:math> , which gives an interesting connection with the discrete Fourier transform and its eigenvectors, as well as all eigenmeasures on $$\\mathbb {R}\\hspace{0.5pt}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>R</mml:mi> <mml:mspace /> </mml:mrow> </mml:math> with uniformly discrete support. An interesting subclass of the latter emerges from the classic cut and project method for aperiodic Meyer sets. Finally, we construct a large class of eigenmeasures with locally finite support that is not uniformly discrete and has large gaps around 0.","PeriodicalId":15993,"journal":{"name":"Journal of Fourier Analysis and Applications","volume":"7 1","pages":"0"},"PeriodicalIF":1.2000,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On Eigenmeasures Under Fourier Transform\",\"authors\":\"Michael Baake, Timo Spindeler, Nicolae Strungaru\",\"doi\":\"10.1007/s00041-023-10045-z\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract Several classes of tempered measures are characterised that are eigenmeasures of the Fourier transform, the latter viewed as a linear operator on (generally unbounded) Radon measures on $$\\\\mathbb {R}\\\\hspace{0.5pt}^d$$ <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\"> <mml:mrow> <mml:mi>R</mml:mi> <mml:msup> <mml:mspace /> <mml:mi>d</mml:mi> </mml:msup> </mml:mrow> </mml:math> . In particular, we classify all periodic eigenmeasures on $$\\\\mathbb {R}\\\\hspace{0.5pt}$$ <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\"> <mml:mrow> <mml:mi>R</mml:mi> <mml:mspace /> </mml:mrow> </mml:math> , which gives an interesting connection with the discrete Fourier transform and its eigenvectors, as well as all eigenmeasures on $$\\\\mathbb {R}\\\\hspace{0.5pt}$$ <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\"> <mml:mrow> <mml:mi>R</mml:mi> <mml:mspace /> </mml:mrow> </mml:math> with uniformly discrete support. An interesting subclass of the latter emerges from the classic cut and project method for aperiodic Meyer sets. Finally, we construct a large class of eigenmeasures with locally finite support that is not uniformly discrete and has large gaps around 0.\",\"PeriodicalId\":15993,\"journal\":{\"name\":\"Journal of Fourier Analysis and Applications\",\"volume\":\"7 1\",\"pages\":\"0\"},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2023-10-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Fourier Analysis and Applications\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1007/s00041-023-10045-z\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Fourier Analysis and Applications","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s00041-023-10045-z","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
摘要
摘要:本文描述了几类调质测度,它们是傅里叶变换的特征测度,后者被看作是$$\mathbb {R}\hspace{0.5pt}^d$$ R d上Radon测度(通常是无界的)的线性算子。特别地,我们对$$\mathbb {R}\hspace{0.5pt}$$ R上的所有周期特征测度进行了分类,它给出了与离散傅里叶变换及其特征向量的有趣联系,以及具有一致离散支持的$$\mathbb {R}\hspace{0.5pt}$$ R上的所有特征测度。后者的一个有趣的子类出现在非周期Meyer集的经典切割和投影方法中。最后,我们构造了一大类具有局部有限支持的特征测度,它不是一致离散的,并且在0附近有很大的间隙。
Abstract Several classes of tempered measures are characterised that are eigenmeasures of the Fourier transform, the latter viewed as a linear operator on (generally unbounded) Radon measures on $$\mathbb {R}\hspace{0.5pt}^d$$ Rd . In particular, we classify all periodic eigenmeasures on $$\mathbb {R}\hspace{0.5pt}$$ R , which gives an interesting connection with the discrete Fourier transform and its eigenvectors, as well as all eigenmeasures on $$\mathbb {R}\hspace{0.5pt}$$ R with uniformly discrete support. An interesting subclass of the latter emerges from the classic cut and project method for aperiodic Meyer sets. Finally, we construct a large class of eigenmeasures with locally finite support that is not uniformly discrete and has large gaps around 0.
期刊介绍:
The Journal of Fourier Analysis and Applications will publish results in Fourier analysis, as well as applicable mathematics having a significant Fourier analytic component. Appropriate manuscripts at the highest research level will be accepted for publication. Because of the extensive, intricate, and fundamental relationship between Fourier analysis and so many other subjects, selected and readable surveys will also be published. These surveys will include historical articles, research tutorials, and expositions of specific topics.
TheJournal of Fourier Analysis and Applications will provide a perspective and means for centralizing and disseminating new information from the vantage point of Fourier analysis. The breadth of Fourier analysis and diversity of its applicability require that each paper should contain a clear and motivated introduction, which is accessible to all of our readers.
Areas of applications include the following:
antenna theory * crystallography * fast algorithms * Gabor theory and applications * image processing * number theory * optics * partial differential equations * prediction theory * radar applications * sampling theory * spectral estimation * speech processing * stochastic processes * time-frequency analysis * time series * tomography * turbulence * uncertainty principles * wavelet theory and applications