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

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.