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Transactions of the American Mathematical Society

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

 
 

 

Necessary conditions for the convergence of cardinal Hermite splines as their degree tends to infinity


Author: T. N. T. Goodman
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
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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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Keywords: Cardinal spline interpolation, cardinal Hermite splines, Euler splines, Chebyshev polynomials
Article copyright: © Copyright 1979 American Mathematical Society