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

Abstract

In the space L(S m−1) of functions integrable on the unit sphere S m−1 of the Euclidean space R m of dimension m ≥ 3, we discuss the problem of one-sided approximation to the characteristic function of a spherical layer G(J) = {x = (x 1 , x 2 ,. .. , xm) ∈ S m−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 L φ (−1, 1) with the ultraspherical weight φ(t) = (1−t 2) 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.