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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Necessary conditions for the convergence of cardinal Hermite splines as their degree tends to infinity
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by T. N. T. Goodman PDF
Trans. Amer. Math. Soc. 255 (1979), 231-241 Request permission

Abstract:

Let ${\mathcal {S}_{n,s}}$ denote the class of cardinal Hermite splines of degree n having knots of multiplicity S at the integers. In this paper we show that if ${f_n} \to f$ uniformly on R, where ${f_n} \in {\mathcal {S}_{{i_{n,s}}}} {i_n} \to \infty$ as $n \to \infty$, and f is bounded, then f is the restriction to R of an entire function of exponential type $\leqslant S$. In proving this result, we need to derive some extremal properties of certain splines ${\mathcal {E}_{n,s}} \in {\mathcal {S}_{n,s}}$, in particular that $||{\mathcal {E}_{n,s}}|{|_\infty }$ minimises $||S|{|_\infty }$ over $S \in {\mathcal {S}_{n,s}}$ with $||{S^{(n)}}|{|_\infty } = ||\mathcal {E}_{n,s}^{(n)}|{|_\infty }$.
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Additional Information
  • © Copyright 1979 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 255 (1979), 231-241
  • MSC: Primary 41A15; Secondary 41A05
  • DOI: https://doi.org/10.1090/S0002-9947-1979-0542878-0
  • MathSciNet review: 542878