Arbitrarily Sparse Spectra for Self-Affine Spectral Measures

IF 0.6 3区数学 Q3 MATHEMATICS

Analysis Mathematica Pub Date : 2023-02-08 DOI:10.1007/s10476-023-0191-9

L.-X. An, C.-K. Lai

引用次数: 7

Abstract

Given an expansive matrix R ∈ M_d(ℤ) and a finite set of digit B taken from ℤ^d/R(ℤ^d). 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.

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自仿射谱测度的任意稀疏谱

给定一个扩张矩阵R∈Md(ℤ) 以及从ℤd/R(ℤd）。先前已经表明，如果我们可以找到一个L，使得（R，B，L）形成Hadamard三元组，那么由（R，B）生成的相关分形自仿射测度允许某个频率集∧的指数正交基，因此它被称为谱测度。在本文中，我们证明了如果#B<det（R）Ş，它不仅是谱的，我们还可以在其Beurling维数为零的意义上构造任意稀疏的谱∧。

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

Analysis Mathematica MATHEMATICS-

CiteScore

1.00

自引率

14.30%

发文量

审稿时长

>12 weeks

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.