Large-scale renormalisation of Fourier transforms of self-similar measures and self-similarity of Riesz measures
type: Item
instance of: scholarly article
title: Large-scale renormalisation of Fourier transforms of self-similar measures and self-similarity of Riesz measures
zbMATH Open document ID: 0849.28006
author: Matthias Holschneider
published in: Journal of Mathematical Analysis and Applications
publication date: 1996-11-10T00:00:00Z
review text: Let \(f\) be a weak limit of \(f_n(\omega)= \prod^{n- 1}_{j= 0} (1+ \cos(3^j \omega))\). It is known as a \(2\pi\)-periodic positive measure. The author shows that in the sense of distributions \(\lim_{n\to \infty} 2^{- n} f(3^{- n}\omega)= f^*(\omega)\) and identifies \(f^*\) as a positive measure that is absolutely continuous with respect to \(f\), \[ {df^*(\omega)\over df(\omega)}= |\widehat \mu(\omega/2)|^2, \] where \(\widehat \mu\) is the Fourier transform of the Cantor measure.
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 42A38
zbMATH DE Number: 897955
zbMATH Keywords: self-similar measure
zbMATH Keywords: Riesz measure
zbMATH Keywords: renormalization
zbMATH Keywords: Fourier transform
zbMATH Keywords: Cantor measure
reviewed by: Hermann Haase
MaRDI profile type: Publication
OpenAlex ID: W1984105242
DOI: 10.1006/JMAA.1996.0206
description: scientific article; zbMATH DE number 897955