Wavelets on Fractals

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 \psi_{1}(x)=\sqrt{2}\chi _{\mathbf{C}}(3x-1) \qquad\text{and}\qquad \psi _{2}(x)=\chi _{\mathbf{C}}(3x)- \chi_{\mathbf{C}}(3x-2) we first describe the subspace in L^{2}(\mathbb{R},(dx)^{s}) which has the following family as an orthonormal basis (ONB): \psi_{i,j,k}(x)=2^{j/2}\psi_{i}(3^{j}x-k)\text{,} 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.