Reconstruction and Decomposition Algorithms for Biorthogonal Multiwavelets

Abstract

We construct biorthogonal multiwavelets (abbreviated to wavelets) in a weighted Hilbert space L^2(E,\rho) where E is a compact subset in \bR^d. A recursive formula for biorthogonal wavelet construction is presented. The construction of the initial wavelets is reformulated as the solution of a certain matrix completion problem. A general solution of the matrix completion problem is identified and expressed conveniently in terms of any given particular solution. Several particular solutions are proposed. Reconstruction and decomposition algorithms are developed for the biorthogonal wavelets. Special results for the univariate case E=[0,1] are obtained.