Hyperinterpolation, introduced by Ian Sloan in %Sloan95, uses a quadrature rule exact on Pi_{2n}, to approximate the coefficients, wrto to an o.n. basis, of the least-squares approximant from Pi_n, thereby obtaining a linear projector onto polynomials of total degree le n that uses only function values and whose L_2 error is within a multiple of the best uniform error achievable by approximants from Pi_n. In more than one variable, the result is not actual interpolation; hence the `hyper'.
%LeGiaSloan01 show that, for L_2 of the unit sphere in RR^r, and with some restriction on the quadrature rule used, the max-norm of the resulting projector grows only like O(n^{r/2-1}).