Uniform approximation on the sphere by least squares polynomials
Provides theoretical guarantees for polynomial approximation on the sphere from scattered data, relevant to geophysics, astrophysics, and signal processing.
The paper proves that least squares polynomial approximation on the sphere achieves optimal Lebesgue constant growth and that delayed arithmetic means of these polynomials converge uniformly at near-best rates, with numerical experiments confirming the theory.
The paper concerns the uniform polynomial approximation of a function $f$, continuous on the unit Euclidean sphere of ${\mathbb R}^3$ and known only at a finite number of points that are somehow uniformly distributed on the sphere. First we focus on least squares polynomial approximation and prove that the related Lebesgue constants w.r.t. the uniform norm grow at the optimal rate. Then, we consider delayed arithmetic means of least squares polynomials whose degrees vary from $n-m$ up to $n+m$, being $m=\lfloor θn\rfloor$ for any fixed parameter $0<θ<1$. As $n$ tends to infinity, we prove that these polynomials uniformly converge to $f$ at the near-best polynomial approximation rate. Moreover, for fixed $n$, by using the same data points we can further improve the approximation by suitably modulating the action ray $m$ determined by the parameter $θ$. Some numerical experiments are given to illustrate the theoretical results.