Cardinal Interpolation and Spline Functions VII. the Behavior of Cardinal Spline Interpolants as their Degree Tends to Infinity

Abstract

Let F(x) be a function from ℝ to ℂ and let

$$S_m \left( x \right)\, = \,\sum\limits_{ - \infty }^\infty {F\left( \nu \right)L_m \left( {x - \nu } \right)}$$

(1)

be the spline function of degree 2m−1, with knots at the integers, that interpolates F(x) at all the integers. We know that S _m (x), as defined by (1), exists if F(x) grows at most like a power | x |^γ as x → ± ∞ (see [7]). Here we investigate conditions which will insure that S _m (x) will converge to F(x) as m → ∞.

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