Self-similarity in harmonic analysis
type: Item
instance of: scholarly article
title: Self-similarity in harmonic analysis
zbMATH Open document ID: 0851.42001
author: Robert S. Strichartz
published in: The Journal of Fourier Analysis and Applications
publication date: 1996-12-02T00:00:00Z
full work available at URL: https://eudml.org/doc/59447
review text: The author surveys his recent work on harmonic analysis and self-similarity. He replaces Lebesgue measure by self-similar measures. The main tool is the Fourier transform of such a measure. For example, the Cantor measure \(\mu = {1\over 2} \mu \circ S^{-1}_0 + {1\over 2} \mu \circ S^{-1}_1\), where \(S_0(x) = {1\over 3} x\) and \(S_1(x) = {1\over 3} x + {2\over 3} ,\) has a Fourier transform \(F\) with the infinite product representation \[ F(\xi) = e^{i\xi/3} \prod^\infty_{k = 1} \cos \xi/3^k. \] He discusses the fractal Plancherel theorem, convolutions, certain dimensions, wavelets, the fractal Radon inversion problem and connections to self-similar tilings of the \(\mathbb{R}^n\). Finally, the results are embedded in the context of stratified nilpotent Lie groups. Many open problems are explicitly mentioned.
Mathematics Subject Classification ID: 42-02
Mathematics Subject Classification ID: 28A80
zbMATH DE Number: 846177
zbMATH Keywords: \(L^ p\) dimensions
zbMATH Keywords: self-similar measures
zbMATH Keywords: Fourier transform
zbMATH Keywords: Plancherel theorem
zbMATH Keywords: wavelets
zbMATH Keywords: Radon inversion problem
zbMATH Keywords: self-similar tilings
reviewed by: Hermann Haase
MaRDI profile type: Publication
OpenAlex ID: W2061127707
DOI: 10.1007/S00041-001-4001-Z
description: scientific article; zbMATH DE number 846177