发表机构
Guangzhou University（广州大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究无平凡无限卷积结构的非均匀自相似测度的谱性，以黄金均值自相似测度为核心例子，结合分析与数值方法发现其傅里叶变换无实零点，为其极可能非谱提供证据，是首个用该框架研究此类测度谱性的工作。
AI 中文摘要
我们研究一类不具有非平凡无限卷积结构的非均匀自相似测度的谱性质，核心例子是黄金均值自相似测度μ，其关联L²空间中指数正交基的存在性是长期悬而未决的公开问题。均匀自相似测度的常用方法在此不适用，需新方法。我们建立该测度的若干基本性质，对其傅里叶变换的零点集开展详细数值研究，采用结合均匀网格采样、二次插值和黄金分割搜索的扫描与细化算法，在宽范围内未发现μ的傅里叶变换的实零点，为μ极可能非谱提供具体证据，表明非均匀性可能是指数正交基存在的自然阻碍。据我们所知，本文是首个通过分析与数值结合框架研究此类测度谱性的尝试。
英文摘要
We investigate the spectral properties of a class of inhomogeneous self-similar measures, which does not admit a non-trivial infinite convolution structure. A central example is the golden-mean self-similar measure $μ$, for which the existence of an exponential orthonormal basis in the associated $L^2$-space has remained a long-standing open problem. The usual approach for homogeneous self-similar measures does not apply here, new methods are required. We establish several basic properties of the measure and then carry out a detailed numerical study of the zero set of its Fourier transform. Using a scanning and refinement algorithm that combines uniform grid sampling, quadratic interpolation, and golden-section search, we examine a wide range and find no real zeros of $\widehatμ$, which provides concrete evidence that $μ$ is very likely non-spectral, suggesting that inhomogeneity may serve as a natural obstruction to the existence of exponential orthonormal bases. To the best of our knowledge, our paper is the first attempt to study the spectrality of such measures through a combined analytic and numerical framework.
