On Filtered Polynomial Approximation on the Sphere

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 $t\ge 2$. 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 $s>d/p$, conditions which in both cases cannot be weakened.