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Approximation order provided by refinable function vectors. (English) Zbl 0870.41015
Summary: In this paper we consider $L_p$-approximation by integer translates of a finite set of functions $\varphi_\nu$ $(\nu=0, \dots, r-1)$ which are not necessarily compactly supported, but have a suitable decay rate. Assuming that the function vector $\varphi= (\varphi_\nu)^{r-1}_{\nu=0}$ is refinable, necessary and sufficient conditions for the refinement mask are derived. In particular, if algebraic polynomials can be exactly reproduced by integer translates of $\varphi_\nu$, then a factorization of the refinement mask of $\varphi$ can be given. This result is a natural generalization of the result for a single function $\varphi$, where the refinement mask of $\varphi$ contains the factor $((1+e^{-iu})/2)^m$ if approximation order $m$ is achieved.

MSC:
41A25Rate of convergence, degree of approximation
41A30Approximation by other special function classes
42A99Fourier analysis in one variable
46C99Inner product spaces, Hilbert spaces
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