On filtered polynomial approximation on the sphere

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 $[0,\mathrm{\infty })$ 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 $\lfloor \frac{d-1}{2}\rfloor$ 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 $W_p^s({\mathbb{S}}^d),1\le p\le \mathrm{\infty }$, both approximations are of the optimal order $L^{-s}$, in the first case for $s>0$ and in the second fully discrete case for $s>d\mathrm{/}p$.