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May, 2006 Wavelets on Fractals
Dorin E. Dutkay , Palle E. T. Jorgensen
Rev. Mat. Iberoamericana 22(1): 131-180 (May, 2006).

Abstract

We show that there are Hilbert spaces constructed from the Hausdorff measures $\mathcal{H}^{s}$ on the real line $\mathbb{R}$ with $0 < s < 1$ which admit multiresolution wavelets. For the case of the middle-third Cantor set $\mathbf{C}\subset \lbrack 0,1]$, the Hilbert space is a separable subspace of $L^{2}(\mathbb{R},(dx)^{s})$ where $s=\log _{3}(2)$. While we develop the general theory of multi-resolutions in fractal Hilbert spaces, the emphasis is on the case of scale $3$ which covers the traditional Cantor set $\mathbf{C}$. Introducing \begin{equation*} \psi_{1}(x)=\sqrt{2}\chi _{\mathbf{C}}(3x-1) \qquad\mbox{and}\qquad \psi _{2}(x)=\chi _{\mathbf{C}}(3x)- \chi_{\mathbf{C}}(3x-2) \end{equation*} we first describe the subspace in $L^{2}(\mathbb{R},(dx)^{s})$ which has the following family as an orthonormal basis (ONB): \begin{equation*} \psi_{i,j,k}(x)=2^{j/2}\psi_{i}(3^{j}x-k)\text{,} \end{equation*} where $i=1,2,j$, $k\in \mathbb{Z}$. Since the affine iteration systems of Cantor type arise from a certain algorithm in $\mathbb{R}^d$ which leaves gaps at each step, our wavelet bases are in a sense gap-filling constructions.

Citation

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Dorin E. Dutkay . Palle E. T. Jorgensen . "Wavelets on Fractals." Rev. Mat. Iberoamericana 22 (1) 131 - 180, May, 2006.

Information

Published: May, 2006
First available in Project Euclid: 24 May 2006

zbMATH: 1104.42021
MathSciNet: MR2268116

Subjects:
Primary: 41A15 , 42A16 , 42A65 , ‎42C40 , ‎43A65 , 46L45 , 46L60 , 47D25

Keywords: Cantor sets , cascade approximation , Fractal , Hausdorff measure , Hilbert space , iterated function systems (IFS) , orthonormal basis (ONB) , scaling , spectrum , Transfer operator , translation , Unitary operators , Wavelets

Rights: Copyright © 2006 Departamento de Matemáticas, Universidad Autónoma de Madrid

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