On filtered polynomial approximation on the sphere

TL;DR

This paper develops and analyzes filtered polynomial approximation methods on the sphere, demonstrating optimal convergence rates for smooth functions using both continuous and fully discrete filtered hyperinterpolation techniques.

Contribution

It introduces new regularity conditions on the filter function and proves optimal approximation orders for both continuous and discrete filtered polynomial approximations on the sphere.

Findings

Optimal approximation order $L^{-s}$ for functions in Sobolev spaces.

Filtered hyperinterpolation achieves the same optimal order as continuous filtering.

Regularity conditions on the filter ensure convergence rates.

Abstract

This paper considers filtered polynomial approximations on the unit sphere $\mathbb{S}^d\subset \mathbb{R}^{d+1}$, obtained by truncating smoothly the Fourier series of an integrable function $f$ with the help of a "filter" $h$, which is a real-valued continuous function on $[0,\infty)$ such that $h(t)=1$ for $t\in[0,1]$ and $h(t)=0$ for $t\ge2$. The resulting "filtered polynomial approximation" (a spherical polynomial of degree $2L-1$) is then made fully discrete by approximating the inner product integrals by an $N$-point cubature rule of suitably high polynomial degree of precision, giving an approximation called "filtered hyperinterpolation". In this paper we require that the filter $h$ and all its derivatives up to $\lfloor\tfrac{d-1}{2}\rfloor$ are absolutely continuous, while its right and left derivatives of order $\lfloor \tfrac{d+1}{2}\rfloor$ exist everywhere and are of bounded variation. Under this assumption we show that for a function $f$ in the Sobolev space $W^s_p(\mathbb{S}^d),\ 1\le p\le \infty$, both approximations are of the optimal order $ L^{-s}$, in the first case for $s>0$ and in the second fully discrete case for $s>d/p$.