POLYNOMIAL INTERPOLATION AND HYPERINTERPOLATION OVER GENERAL REGIONS

This paper studies a generalization of polynomial interpolation: given a continuous function over a rather general manifold, hyperinterpolat ion is a linear approximation that makes use of values off on a well c hosen finite set. The approximation is a discrete least-squares approx imation constructed with the aid of a high-order quadrature rule: the role of the quadrature rule is to approximate the Fourier coefficients off with respect to an orthonormal basis of the space of polynomials of degree less than or equal to n. The principal result is a generaliz ation of the result of Erdos and Turan for classical interpolation at the zeros of orthogonal polynomials: for a rule of suitably high order (namely 2n or greater), the L(2) error of the approximation is shown to be within a constant factor of the error of best uniform approximat ion by polynomials of degree less than or equal to n. The L(2) error t herefore converges to zero as the degree of the approximating polynomi al approaches co. 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 greater than or equal to 3. In su ch a situation the classical interpolation property cannot hold, where as satisfactory hyperinterpolation approximations do exist. (C) 1995 A cademic Press, Inc.