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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 $ \vert\vert{\mathcal{E}_{n,s}}\vert{\vert _\infty }$ minimises $ \vert\vert S\vert{\vert _\infty }$ over $ S\, \in \,{\mathcal{S}_{n,s}}$ with $ \vert\vert{S^{(n)}}\vert{\vert _\infty }\, = \,\vert\vert\mathcal{E}_{n,s}^{(n)}\vert{\vert _\infty }$.


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DOI: https://doi.org/10.1090/S0002-9947-1979-0542878-0
Keywords: Cardinal spline interpolation, cardinal Hermite splines, Euler splines, Chebyshev polynomials
Article copyright: © Copyright 1979 American Mathematical Society