TY - JOUR

T1 - ONE-SIDED L-APPROXIMATION ON A SPHEREOF THE CHARACTERISTIC FUNCTION OF A LAYER

AU - Deikalova, Marina V.

AU - Torgashova, Anastasiya Yu.

PY - 2018

Y1 - 2018

N2 - 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.

AB - 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.

UR - http://elibrary.ru/item.asp?id=36702169

U2 - 10.15826/umj.2018.2.003

DO - 10.15826/umj.2018.2.003

M3 - Article

VL - 4

SP - 13

EP - 23

JO - Ural Mathematical Journal

JF - Ural Mathematical Journal

SN - 2414-3952

IS - 2 (7)

ER -