Gerlind Plonka
Approximation Order of Shift-Invariant Subspaces of $L^2(\Bbb{R})$ Generated by Refinable Function Vectors
Preprint series: Preprints aus dem Fachbereich Mathematik, Universität Rostock

MSC:

41A25 Rate of convergence, degree of approximation

41A30 Approximation by other special function classes

Abstract: In this paper, finitely generated shift-invariant subspaces $S(\phi)$
of $L^2(\Bbb{R})$ providing controlled approximatin order $m$ are
considered. Assuming that the vector of generating functins $\phi$ of
$S(\phi)$ is refinable, necessary and sufficient conditions for the
refinement mask are derived. In particular, if algebraic polynomials
can be exactly reproduced in $S(\phi)$, then a factorization of the
refinement mask can be given. This result is a natural generalization
of the result of the principal shift-invariant subspaces of $L^2(\Bbb{R})$
where the refinement mask contains a power of $\frac{1+e^{-iu}}{2}$
if the scaling function is regular.
Keywords: finitely generated subspaces of $L^2(\Bbb{R})$, controlled approximation order, accuracy refinement mask, Strang-Fix conditions