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Regularity of refinable function vectors. (English) Zbl 0914.42025
Compactly supported solutions of refinement equations \(\phi(x)=\sum_{n=0}^N P_n\phi(2x-n)\) are considered for vector valued functions \(\phi\) and constant square matrices \(P_n\). As in the scalar case, the smoothness of \(\phi\) and the decay of the Fourier transform \(\widehat{\phi}\) are characterized by properties of the refinement mask \(P(u)={1\over 2}\sum_n P_ne^{-inu}\). In the scalar case, the approximation order can only be \(m\) if \(P(u)\) factors as \([(1+e^{-iu})/2]^mP^{(m)}(u)\) where \(P^{(m)}\) is \(2\pi\)-periodic, and \(P^{(m)}(0)=1\). The factorization property needed in the vector case is however much more involved: a factorization of the form \[ P(u)=2^{-m}C_0(2u)\cdots C_{m-1}(2u) P^{(m)}(u) C_{m-1}(u)^{-1}\cdots C_0(u)^{-1} \] is needed. From the relation \(\widehat{\phi}(u)=P(u/2)\widehat{\phi}(u/2)\), one gets the well-known infinite product representation of \(\widehat{\phi}\). The convergence of the infinite product in the matrix case is again much more intricate than in the scalar case and needs more conditions on \(P(u)\), for example \(P(0)\) should be diagonalizable with spectral radius at most 1. Finally, some conditions are needed to make \(\widehat{\phi}(u)\) decay for \(| u| \to\infty\). These decay properties allow to prove uniqueness of the solution of the refinement equations in a large function class and also the convergence of the cascade and subdivision algorithm can be derived. Several examples illustrate these results. The scaling functions of G. C. Donovan, J. S. Geronimo, D. P. Hardin and P. R. Massopust [SIAM J. Math. Anal. 27, No. 4, 1158-1192 (1996; Zbl 0873.42021)] are a special case.

MSC:
42C15 General harmonic expansions, frames
39B62 Functional inequalities, including subadditivity, convexity, etc.
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
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