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Solution of two problems on wavelets

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Abstract

We solve two problems on wavelets. The first is the nonexistence of a regular wavelet that generates a wavelet basis for the Hardy space ℍ2(ℝ). The second is the existence, given any regular wavelet basis for\(\mathbb{H}^2 (\mathbb{R})\), of aMulti-Resolution Analysis generating the wavelet. Moreover, we construct a regular scaling function for this Multi-Resolution Analysis. The needed regularity conditions are very mild and our proofs apply to both the orthonormal and biorthogonal situations. Extensions to more general cases in dimension 1 and higher are given. In particular, we show in dimension larger than 2 that a regular wavelet basis for\(\mathbb{L}^2 (\mathbb{R}^n )\) arises from a Multi-Resolution Analysis that is regular modulo the action of a unitary operator, which is whenn = 2 a Calderón-Zygmund operator of convolution type.

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Communicated by Guido Weiss

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Auscher, P. Solution of two problems on wavelets. J Geom Anal 5, 181–236 (1995). https://doi.org/10.1007/BF02921675

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