One-sided $L$-approximation on a sphere of the characteristic function of a layer

In the space $L(\mathbb{S}^{m-1})$ of functions integrable on the unit sphere $\mathbb{S}^{m-1}$ of the Euclidean space $\mathbb{R}^{m}$ of dimension $m\ge 3$, we discuss the problem of one-sided approximation to the characteristic function of a spherical layer $\mathbb{G}(J)=\{x=(x_1,x_2,\ldots,x_m)\in \mathbb{S}^{m-1}\colon x_m\in J\},$ where $J$ is one of the intervals $(a,1],$ $(a,b),$ and $[-1,b),$ $-1< a<b< 1,$ by the set of algebraic polynomials of given degree $n$ in $m$ variables. This problem reduces to the one-dimensional problem of one-sided approximation in the space $L^\phi(-1,1)$ with the ultraspherical weight $ \phi(t)=(1-t^2)^\alpha,\ \alpha=(m-3)/2$, to the characteristic function of the interval $J$. This result gives a solution of the problem of one-sided approximation to the characteristic function of a spherical layer in all cases when a solution of the corresponding one-dimensional problem known. In the present paper, we use results by A.G. Babenko, M.V. Deikalova, and Sz.G. Revesz (2015) and M.V. Deikalova and A.Yu. Torgashova (2018) on the one-sided approximation to the characteristic functions of intervals.