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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.

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