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Approximation order of two-direction multiscaling functions. (English) Zbl 1436.42035

A matrix \(A\) satisfies condition E if it has a simple eigenvalue of 1, and all other eigenvalues are smaller than 1 in absolute value. Condition E for two-direction multiscaling functions \(\phi\) is necessary for the stability of the multiresolution approximation produced by \(\phi\) or for the existence of \(\phi\) in the matrix refinement equation of \(\phi\). Condition E for \(\Phi(x) := [\phi(x);\phi(-x)]^{T}\) is used as an equivalent condition for condition E for \(\phi\). In this article the author obtains a simple and efficient criterion for condition E for \(\phi\), criteria for the basic regularity conditions for \(\phi\) as well as a criterion for approximation order of \(\phi\).
The author also gives examples for illustrating the general theory.

MSC:

42C15 General harmonic expansions, frames
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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References:

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