ONE-SIDED \(L\)-APPROXIMATION ON A SPHERE OF THE CHARACTERISTIC FUNCTION OF A LAYER

Abstract

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.