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

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摘要

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.
源语言English
页(从-至)13-23
期刊Ural Mathematical Journal
4
2 (7)
DOI
Published - 2018

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  • 27.00.00 MATHEMATICS

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