One-Sided L-Approximation on a Sphere of the Characteristic Function of a Layer

Abstract

In the space L(Sm−1) of functions integrable on the unit sphere Sm−1 of the Euclidean space Rm of dimension m≥3, we discuss the problem of one-sided approximation to the characteristic function of a spherical layer G(J)={x=(x1,x2,…,xm)∈Sm−1:xm∈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 Lϕ(−1,1) with the ultraspherical weight ϕ(t)=(1−t2)α, α=(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.