Biorthogonal bases of compactly supported wavelets. (English) Zbl 0776.42020

A compactly supported wavelet, which generates an orthonormal basis of \(L^ 2(-\infty,\infty)\), cannot be symmetric or anti-symmetric, unless it is some integer-translate and modulation by \(\pm 1\) of the Haar function. On the other hand, a compactly supported symmetric (or anti- symmetric) wavelet, which generates a semi-orthogonal basis of \(L^ 2(- \infty,\infty)\) in the sense that orthogonality is achieved only among different scales, does not have a compactly supported dual. The objective of this fundamental paper is to describe the structure, give a detailed analysis, and provide several examples, of a pair of compactly supported symmetric (or anti-symmetric) dual wavelets (or biorthogonal wavelets), with arbitrary desirable regularity. Hence, instead of one multiresolution analysis, two nested sequences of multiresolution analyses are considered. It is also shown how to construct compactly supported symmetric biorthogonal wavelets that are close to a non- symmetric orthonormal one.


42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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