A formula for the error of finite sinc-interpolation over a finite interval

Abstract

Sinc-interpolation is a very efficient infinitely differentiable approximation scheme from equidistant data on the infinite line. It, however, requires that the interpolated function decreases rapidly or is periodic. We give an error formula for the case where neither of these conditions is satisfied.

A formula for the error of finite sinc interpolation with an even number of nodes

“…We shall consider a fixed interval [−X, X], X ∈ IR + , and at first choose as in [Ber3] some h such that X = Nh for N ∈ IN N to approximate C(f, h) of (1.1) with the finite interpolant 2) where the double prime denotes that the first and last terms are halved. This permits to write C N as a function of the difference of two classical quadrature formulae, see §3.…”

“…This permits to write C N as a function of the difference of two classical quadrature formulae, see §3. We are interested in the error C N (f, h) − f as a function of h for f ∈ C q [−X, X] for some q ∈ IN N. After completing the error term for a formula given in [Ber3], we derive in §4 the corresponding one for an even number of x n .…”

“…As in [Ber3], our analysis rests on the study of errors in numerical quadrature. We shall use the notation I [a,b] := b a f (y)dy for the usual definite integral and I [a,b] x := PV b a f (y)…”

First applications of a formula for the error of finite sinc interpolation

“…(M h := midpoint rule approximation to I x with the steplength h [7,8]). The jump in f on the circle now is at 0 ≡ X and the error formula is the same as (1.3) with (−1) M replaced by 1 and −X by 0 as argument of the last derivative.…”

“…For example, one has C( f, h)(x) = f (x) ∀ x when f belongs to the Paley-Wiener class with exponent π/h [19,21]; also, C( f, h)(x) converges exponentially toward f (x) when f is analytic in a symmetric strip about the real axis and decays fast enough on the lines that make up the boundary of the strip [19,21]. One obviously has [7] C…”