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A characterization of wavelet families arising from biorthogonal MRA’s of multiplicityd

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Abstract

In this article we give a necessary and sufficient condition for a pair of wavelet families\(\Psi = \{ \psi ^1 ,...,\psi ^L \} , \tilde \Psi = \{ \tilde \psi ^1 ,...,\tilde \psi ^L \} \) in L2(ℝn), to arise from a pair of biorthogonal MRA’s. The condition is given in terms of simple equations involving the functions ψℓ and\(\tilde \psi ^\ell \). To work in greater generality, we allow multiresolution analyses of arbitrary multiplicity, based on lattice translations and matrix dilations. Our result extends the characterization theorem of G. Gripenberg and X. Wang for dyadic orthonormal wavelets in L2(ℝ),and includes, as particular cases, the sufficient conditions of P. Auscher and P.G. Lemarié in the biorthogonal situation.

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Authors and Affiliations

  1. Dipartimento di Statistica, Université di Milano-Bicocca, Via Bicocca degli Arcimboldi 8, 20126, Milano, Italy

    Andrea Calogero

  2. Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129, Torino, Italy

    Gustavo Garrigós

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  1. Andrea Calogero
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  2. Gustavo Garrigós
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Correspondence to Andrea Calogero.

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Communicated by Guido Weiss

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Calogero, A., Garrigós, G. A characterization of wavelet families arising from biorthogonal MRA’s of multiplicityd . J Geom Anal 11, 187–217 (2001). https://doi.org/10.1007/BF02921962

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  • DOI: https://doi.org/10.1007/BF02921962

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