We show that there are Hilbert spaces constructed from the Hausdorff measures on the real line with which admit multiresolution wavelets. For the case of the middle-third Cantor set , the Hilbert space is a separable subspace of where . While we develop the general theory of multi-resolutions in fractal Hilbert spaces, the emphasis is on the case of scale which covers the traditional Cantor set . Introducing
we first describe the subspace in which has the following family as an orthonormal basis (ONB):
where , . Since the affine iteration systems of Cantor type arise from a certain algorithm in which leaves gaps at each step, our wavelet bases are in a sense gap-filling constructions.
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Dorin E. Dutkay, Palle E.T. Jorgensen, Wavelets on Fractals. Rev. Mat. Iberoam. 22 (2006), no. 1, pp. 131–180DOI 10.4171/RMI/452