Euler-Maclaurin and Gregory interpolants

Let a sufficiently smooth function f on [−1, 1] be sampled at n + 1 equispaced points, and let k ≥ 0 be given. An Euler–Maclaurin interpolant to the data is defined, consisting of a sum of a degree k algebraic polynomial and a degree n trigonometric polynomial, which deviates from f by O(n−k) and whose integral is equal to the order k Euler–Maclaurin approximation of the integral of f . This interpolant makes use of the same derivatives f ( j)(±1) as the Euler–Maclaurin formula. A variant Gregory interpolant is also defined, based on finite difference approximations to the derivatives, whose integral (for k odd) is equal to the order k Gregory approximation to the integral. Mathematics Subject Classification 41A05 · 42A15 · 65D32 · 65D05