On the Remainders and the Convergence of Cardinal Spline Interpolation for Almost Periodic Functions

Abstract

Remainders in terms of high-order derivatives might at times seem rather useless for numerical applications. However, they are often effective in theoretical problems of convergence. Our present topic is the remainder of cardinal spline interpolation (C.S.I.) of the odd degree 2m-1, in its customary Peano form. Its kernel K_2m-1(x,t) appears to be endowed with numerous worthwhile properties some of which are described in the first half (Part I) of this paper. The remainder of C.S.I. allows us to discuss the behavior of the interpolant, as m → ∞, of entire functions of exponential type (Theorem 6 of §6).