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

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(ℝ n ), 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. Lemarie in the biorthogonal situation.