# 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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## Cardinal Hermite spline interpolation: convergence as the degree tends to infinityHTML articles powered by AMS MathViewer

by M. J. Marsden and S. D. Riemenschneider
Trans. Amer. Math. Soc. 235 (1978), 221-244 Request permission

## Abstract:

Let ${\mathcal {S}_{2m,r}}$, denote the class of cardinal Hermite splines of degree $2m - 1$ having knots of multiplicity r at the integers. For $f(x) \in {C^{r - 1}}(R)$, the cardinal Hermite spline interpolant to $f(x)$ is the unique element of ${\mathcal {S}_{2m,r}}$ which interpolates $f(x)$ and its first $r - 1$ derivatives at the integers. For $y = ({y^0}, \ldots ,{y^{r - 1}})$ an r-tuple of doubly-infinite sequences, the cardinal Hermite spline interpolant to y is the unique $S(x) \in {\mathcal {S}_{2m,r}}$ satisfying ${S^{(s)}}(\nu ) = {y^s},s = 0,1, \ldots ,r - 1$, and $\nu$ an integer. The following results are proved: If $f(x)$ is a function of exponential type less than $r\pi$, then the derivatives of the cardinal Hermite spline interpolants to $f(x)$ converge uniformly to the respective derivatives of $f(x)$ as $m \to \infty$. For functions from more general, but related, classes, weaker results hold. If y is an r-tuple of ${l^p}$ sequences, then the cardinal Hermite spline interpolants to y converge to ${W_r}(y)$, a certain generalization of the Whittaker cardinal series which lies in the Sobolev space ${W^{p,r - 1}}(R)$. This convergence is in the Sobolev norm. The class of all such ${W_r}(y)$ is characterized. For small values of r, the explicit forms of ${W_r}(y)$ are described.
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