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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Convergence of cardinal series

Authors: Carl de Boor, Klaus Höllig and Sherman Riemenschneider
Journal: Proc. Amer. Math. Soc. 98 (1986), 457-460
MSC: Primary 41A30
MathSciNet review: 857940
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Abstract: The result of this paper is a generalization of our characterization of the limits of multivariate cardinal splines. Let $ {M_n}$ denote the $ n$-fold convolution of a compactly supported function $ M \in {L_2}({{\mathbf{R}}^d})$ and denote by

$\displaystyle {S_n}: = \left\{ {\sum\limits_{j \in {{\mathbf{Z}}^d}} {c(j){M_n}( \cdot - j):c \in {l_2}({{\mathbf{Z}}^d})} } \right\}$

the span of the translates of $ {M_n}$. We prove that there exists a set $ \Omega $ with $ {\operatorname{vol} _d}(\Omega ) = {(2\pi )^d}$ such that for any $ f \in {L_2}({{\mathbf{R}}^d})$,

$\displaystyle \operatorname{dist} (f,{S_n}) \to 0\quad {\text{as }}n \to \infty ,$

if and only if the support of the Fourier transform of $ f$ is contained in $ \Omega $.

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Keywords: Multivariate, cardinal series, convergence, Fourier transform
Article copyright: © Copyright 1986 American Mathematical Society