Pairs of dual wavelet frames from any two refinable functions. (English) Zbl 1055.42025

A function \(\phi\in L^2(R)\) is \(d\)-refinable if \(\phi= \sum_{k\in Z} a_k\phi(d \cdot -k)\) for some (finite) scalar sequence \(\{a_k\}\). Given two compactly supported \(d\)-refinable functions, it is shown how to construct \(2d\) compactly supported functions \(\{\psi_1, \dots, \psi_d\}\) and \(\{\tilde{\psi}_1, \dots, \tilde{\psi}_d\}\) generating a pair of dual wavelet frames for \(L^2(R)\), with the maximal number of vanishing moments. The functions \(\psi_j,\tilde{\psi}_j\) are finite linear combinations of the translates \(\phi(d \cdot -k), k\in Z\). The results are applied to construct pairs of dual frames generated from B-splines.


42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
42C15 General harmonic expansions, frames
41A15 Spline approximation
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