## 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