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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Convergence of cardinal series
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by Carl de Boor, Klaus Höllig and Sherman Riemenschneider PDF
Proc. Amer. Math. Soc. 98 (1986), 457-460 Request permission

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 \[ {S_n}: = \left \{ {\sum \limits _{j \in {{\mathbf {Z}}^d}} {c(j){M_n}( \cdot - j):c \in {l_2}({{\mathbf {Z}}^d})} } \right \}\] 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 $f \in {L_2}({{\mathbf {R}}^d})$, \[ \operatorname {dist} (f,{S_n}) \to 0\quad {\text {as }}n \to \infty ,\] if and only if the support of the Fourier transform of $f$ is contained in $\Omega$.
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Additional Information
  • © Copyright 1986 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 98 (1986), 457-460
  • MSC: Primary 41A30
  • DOI: https://doi.org/10.1090/S0002-9939-1986-0857940-1
  • MathSciNet review: 857940