Polynomial Interpolation and Hyperinterpolation over General Regions

AbstractThis 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 2n or greater), the L2 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 L2 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 Rs 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