Wavelets on fractals and Besov spaces
type: Item
instance of: scholarly article
title: Wavelets on fractals and Besov spaces
zbMATH Open document ID: 0912.42025
DOI: 10.1007/BF02476031
author: Alf Jonsson
published in: The Journal of Fourier Analysis and Applications
publication date: 1999-04-27T00:00:00Z
full work available at URL: https://eudml.org/doc/59568
review text: The author develops the theory of wavelets on self-similar fractals \(K\) which preserves a certain standard inequality (Markov's inequality). This property is equivalent to the statement that for all balls \(B\) centered at a point in \(K\), and all polynomials \(P\) of degree at most \(m\), one has \[ \sup_{x \in B} | P(x)| \leq C_m \sup_{x \in B \cap K} | P(x)| . \] For such a fractal \(K\) and a positive integer \(m\), an orthonormal wavelet basis is constructed; the basis elements are compositions of piecewise polynomial functions of degree at most \(m\) with the similitudes of \(K\). This wavelet basis is then shown to characterise the Besov spaces \(B^{p,q}_\alpha(K)\) in the expected manner. In particular, if \(\dim(K) = s\) and \(f\) is a function in the Lipschitz space \(\Lambda_\alpha(K)\) for some \(\alpha > 0\), then the wavelet coefficients at scale \(i\) decay like \(\text{diam} (K_i)^{\alpha+s/2}\) providing that \(m \geq {\alpha}\), where \(i\) is a sequence of similitudes and \(K_i\) is the image of \(K\) under this sequence. The converse is also true if the \(K_i\) are disjoint; this statement does not require \(m \geq {\alpha}\).
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 28A80
zbMATH DE Number: 1220781
zbMATH Keywords: wavelets
zbMATH Keywords: function spaces
zbMATH Keywords: fractals
zbMATH Keywords: Besov spaces
reviewed by: Terence Tao
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1220781