Polynomial Interpolation and Hyperinterpolation over General Regions

Abstract

This paper studies a generalization of polynomial interpolation: given a continuous function over a rather general manifold, hyperinterpolation is a linear approximation that makes use of values of f on a well chosen finite set. The approximation is a discrete least-squares approximation constructed with the aid of a high-order quadrature rule: the role of the quadrature rule is to approximate the Fourier coefficients of f with respect to an orthonormal basis of the space of polynomials of degree ≤ n. The principal result is a generalization of the result of Erdös and Turan for classical interpolation at the zeros of orthogonal polynomials: for a rule of suitably high order (namely 2 n or greater), the L 2 error of the approximation is shown to be within a constant factor of the error of best uniform approximation by polynomials of degree ≤ n. The L 2 error therefore converges to zero as the degree of the approximating polynomial approaches ∞. An example discussed in detail is the approximation of continuous functions on the sphere in R s by spherical polynomials. In this case the number of quadrature points must exceed the number of degrees of freedom if n > 2 and s ≥ 3. In such a situation the classical interpolation property cannot hold, whereas satisfactory hyperinterpolation approximations do exist.