Euler–Maclaurin and Gregory interpolants

Let a sufficiently smooth function $$f$$f on $$[-1,1]$$[-1,1] be sampled at $$n+1$$n+1 equispaced points, and let $$k\ge 0$$k≥0 be given. An Euler–Maclaurin interpolant to the data is defined, consisting of a sum of a degree $$k$$k algebraic polynomial and a degree $$n$$n trigonometric polynomial, which deviates from $$f$$f by $$O(n^{-k})$$O(n-k) and whose integral is equal to the order $$k$$k Euler–Maclaurin approximation of the integral of $$f$$f. This interpolant makes use of the same derivatives $$f^{( j)}(\pm 1)$$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 $$k$$k odd) is equal to the order $$k$$k Gregory approximation to the integral.