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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 ${\varphi }_{\nu }$ $\left(\nu =0,\cdots ,r-1\right)$ which are not necessarily compactly supported, but have a suitable decay rate. Assuming that the function vector $\varphi ={\left({\varphi }_{\nu }\right)}_{\nu =0}^{r-1}$ 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 ${\left(\left(1+{e}^{-iu}\right)/2\right)}^{m}$ if approximation order $m$ is achieved.

##### MSC:
 41A25 Rate of convergence, degree of approximation 41A30 Approximation by other special function classes 42A99 Fourier analysis in one variable 46C99 Inner product spaces, Hilbert spaces