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Polynomial interpolation and hyperinterpolation over general regions. (English) Zbl 0839.41006

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 \(\leq n\). The principal result is a generalization of the result of Erdös and Turán 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 approximation by polynomials of degree \(\leq n\). The \(L_2\) error therefore converges to zero as the degree of the approximating polynomial approach \(\infty\). As example discussed in detail is the approximation of continuous functions on the sphere in \(\mathbb{R}^s\) by spherical polynomials.

MSC:

41A10 Approximation by polynomials
41A55 Approximate quadratures
65D32 Numerical quadrature and cubature formulas
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