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Cardinal Hermite spline interpolation: convergence as the degree tends to infinity


Authors: M. J. Marsden and S. D. Riemenschneider
Journal: Trans. Amer. Math. Soc. 235 (1978), 221-244
MSC: Primary 41A05
DOI: https://doi.org/10.1090/S0002-9947-1978-0463752-3
MathSciNet review: 0463752
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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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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1978-0463752-3
Keywords: Cardinal Hermite splines, Hermite interpolation, Whittaker series, summability, Sobolev spaces
Article copyright: © Copyright 1978 American Mathematical Society

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