## 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