Biorthogonal bases of compactly supported wavelets

TLDR

Orthonormal compactly supported wavelet bases correspond to subband coding schemes with exact reconstruction where analysis and synthesis filters coincide. The authors derive necessary and sufficient conditions for biorthogonality of the scaling functions, provide sufficient conditions for decay of their Fourier transforms, and analyze the regularity of the resulting biorthogonal bases. Under general conditions, exact reconstruction schemes with distinct synthesis filters yield dual Riesz bases of compactly supported wavelets; the authors present symmetric families with arbitrarily high regularity and constructions close to nonsymmetric orthonormal bases.

Abstract

Abstract Orthonormal bases of compactly supported wavelet bases correspond to subband coding schemes with exact reconstruction in which the analysis and synthesis filters coincide. We show here that under fairly general conditions, exact reconstruction schemes with synthesis filters different from the analysis filters give rise to two dual Riesz bases of compactly supported wavelets. We give necessary and sufficient conditions for biorthogonality of the corresponding scaling functions, and we present a sufficient conditions for the decay of their Fourier transforms. We study the regularity of these biorthogonal bases. We provide several families of examples, all symmetric (corresponding to “linear phase” filters). In particular we can construct symmetric biorthogonal wavelet bases with arbitraily high preassigned regularity; we also show how to construct symmetric biorthogonal wavelet bases “close” to a (nonsymmetric) orthonormal basis.