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.

Keywords

Biorthogonal systemOrthonormal basisMathematicsWaveletBiorthogonal waveletMultiresolution analysisPure mathematicsBasis (linear algebra)Mathematical analysisWavelet transformDiscrete wavelet transformComputer scienceGeometry

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Publication Info

Year
1992
Type
article
Volume
45
Issue
5
Pages
485-560
Citations
2618
Access
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Cite This

Albert Cohen, Ingrid Daubechies, Jean-Christophe Feauveau (1992). Biorthogonal bases of compactly supported wavelets. Communications on Pure and Applied Mathematics , 45 (5) , 485-560. https://doi.org/10.1002/cpa.3160450502

Identifiers

DOI
10.1002/cpa.3160450502