Construction of a class of spectral measures
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.
scientific article; zbMATH DE number 7612314
scientific article