{"title":"自仿射谱测度的任意稀疏谱","authors":"L.-X. An, C.-K. Lai","doi":"10.1007/s10476-023-0191-9","DOIUrl":null,"url":null,"abstract":"<div><p>Given an expansive matrix <i>R</i> ∈ <i>M</i><sub><i>d</i></sub>(ℤ) and a finite set of digit <i>B</i> taken from ℤ<sup><i>d</i></sup>/<i>R</i>(<i>ℤ</i><sup><i>d</i></sup>). It was shown previously that if we can find an <i>L</i> such that (<i>R, B, L</i>) forms a Hadamard triple, then the associated fractal self-affine measure generated by (<i>R, B</i>) admits an exponential orthonormal basis of certain frequency set Λ, and hence it is termed as a spectral measure. In this paper, we show that if #<i>B</i> < ∣det(<i>R</i>)∣, not only it is spectral, we can also construct arbitrarily sparse spectrum Λ in the sense that its Beurling dimension is zero.</p></div>","PeriodicalId":55518,"journal":{"name":"Analysis Mathematica","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2023-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"7","resultStr":"{\"title\":\"Arbitrarily Sparse Spectra for Self-Affine Spectral Measures\",\"authors\":\"L.-X. An, C.-K. Lai\",\"doi\":\"10.1007/s10476-023-0191-9\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Given an expansive matrix <i>R</i> ∈ <i>M</i><sub><i>d</i></sub>(ℤ) and a finite set of digit <i>B</i> taken from ℤ<sup><i>d</i></sup>/<i>R</i>(<i>ℤ</i><sup><i>d</i></sup>). It was shown previously that if we can find an <i>L</i> such that (<i>R, B, L</i>) forms a Hadamard triple, then the associated fractal self-affine measure generated by (<i>R, B</i>) admits an exponential orthonormal basis of certain frequency set Λ, and hence it is termed as a spectral measure. In this paper, we show that if #<i>B</i> < ∣det(<i>R</i>)∣, not only it is spectral, we can also construct arbitrarily sparse spectrum Λ in the sense that its Beurling dimension is zero.</p></div>\",\"PeriodicalId\":55518,\"journal\":{\"name\":\"Analysis Mathematica\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2023-02-08\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"7\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Analysis Mathematica\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s10476-023-0191-9\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Analysis Mathematica","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10476-023-0191-9","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
Arbitrarily Sparse Spectra for Self-Affine Spectral Measures
Given an expansive matrix R ∈ Md(ℤ) and a finite set of digit B taken from ℤd/R(ℤd). It was shown previously that if we can find an L such that (R, B, L) forms a Hadamard triple, then the associated fractal self-affine measure generated by (R, B) admits an exponential orthonormal basis of certain frequency set Λ, and hence it is termed as a spectral measure. In this paper, we show that if #B < ∣det(R)∣, not only it is spectral, we can also construct arbitrarily sparse spectrum Λ in the sense that its Beurling dimension is zero.
期刊介绍：
Traditionally the emphasis of Analysis Mathematica is classical analysis, including real functions (MSC 2010: 26xx), measure and integration (28xx), functions of a complex variable (30xx), special functions (33xx), sequences, series, summability (40xx), approximations and expansions (41xx).
The scope also includes potential theory (31xx), several complex variables and analytic spaces (32xx), harmonic analysis on Euclidean spaces (42xx), abstract harmonic analysis (43xx).
The journal willingly considers papers in difference and functional equations (39xx), functional analysis (46xx), operator theory (47xx), analysis on topological groups and metric spaces, matrix analysis, discrete versions of topics in analysis, convex and geometric analysis and the interplay between geometry and analysis.
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