Cardinal Interpolation and Spline Functions IV. The Exponential Euler Splines

Abstract

As background for our discussion we recall a result from [10]. First a few definitions. Let ℐ _n denote the class of cardinal spline functions S(x) of degree n (n≧1) having their knots at the integer points of the real axis. This means that S(x)∈ℐ _n , provided that the restriction of S(x) to every unit interval (v, v + 1) is a polynomial of degree n at most, and that

$$S\left( x \right) \in C^{n - 1} \left( { - \infty ,\,\infty } \right).$$

(1)

Furthermore, let

$$\vartheta _n^* \, = \,\left\{ {S\left( x \right);\,S\left( {x + \frac{1} {2}} \right) \in \vartheta _n } \right\}.$$

(2)

The elements of ℐ* _n are again cardinal spline functions of degree n, but having their knots at v+1/2 half way between consecutive integers.

Sponsored by the U. S. Army under Contract No. DA—31–124—ARO—D—462.