Construction of a class of spectral measures
type: Item
instance of: scholarly article
title: Construction of a class of spectral measures
zbMATH Open document ID: 1504.28014
author: Hai-Hua Wu
author: Jing-Cheng Liu
published in: Forum Mathematicum
publication date: 2022-11-04T00:00:00Z
review text: One can begin with authors' abstract: ``Let the iterated function systems \(\{S_i\}^N _{i=1}\) be defined by \(S_i(x)=(-1)^{i-1}\rho (x+d_i)\), \(x\in \mathbb R\), \(d_i \in D\), where \(0<\rho<1\), and \(D\) is a finite subset of \(\mathbb Z\). Let the measure \(\mu_{\rho, P}\) be generated by the IFS and the probability weight \(P=\{p_i\}^N _{i=1}\). In this paper, we introduce a new way to consider the spectrality of \(\mu_{\rho, P}\), and obtain a sufficient and necessary condition for \(N=2\), and construct two examples of spectral measures for \(N=2L \ge 4\) and \(N=2L+1\), respectively.'' In the survey of this paper, the attention is given to Fuglede's spectral set conjecture and to the difference in the spectral theory among absolutely continuous measures, singularly continuous measures, and counting measures. Also, such notions as a spectral measure, a spectrum of the measure, and a self-similar measure, as well as a self-affine measure, a spectral set, and an attractor, the Fourier transform of a Borel probability measure, etc., are described. Finally, some open problems and examples of spectral measures are noted.
reviewed by: Symon Serbenyuk
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 42C05
zbMATH DE Number: 7612314
zbMATH Keywords: Fourier transform
zbMATH Keywords: spectrum
zbMATH Keywords: spectral measure
zbMATH Keywords: self-affine measure
zbMATH Keywords: compatible pair
MaRDI profile type: Publication
full work available at URL: forum-2022-0011
OpenAlex ID: W4294094065
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DOI: 10.1515/FORUM-2022-0011
description: scientific article; zbMATH DE number 7612314