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

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 $||{S^{(n)}}|{|_\infty } = ||\mathcal {E}_{n,s}^{(n)}|{|_\infty }$.