Polynomial approximation on the sphere using scattered data

We consider the problem of approximately reconstructing a function f defined on the surface of the unit sphere in the Euclidean space ℝq +1 by using samples of f at scattered sites. A central role is played by the construction of a new operator for polynomial approximation, which is a uniformly bounded quasi-projection in the de la Vallee Poussin style, i.e. it reproduces spherical polynomials up to a certain degree and has uniformly bounded Lp operator norm for 1 ≤ p ≤ ∞. Using certain positive quadrature rules for scattered sites due to Mhaskar, Narcowich and Ward, we discretize this operator obtaining a polynomial approximation of the target function which can be computed from scattered data and provides the same approximation degree of the best polynomial approximation. To establish the error estimates we use Marcinkiewicz–Zygmund inequalities, which we derive from our continuous approximating operator. We give concrete bounds for all constants in the Marcinkiewicz–Zygmund inequalities as well as in the error estimates. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)