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

Marina V. Deikalova     (Ural Federal University, 51 Lenin aven., Ekaterinburg, 620000, Russian Federation)
Anastasiya Yu. Torgashova     (Ural Federal University, 51 Lenin aven., Ekaterinburg, 620000, Russian Federation)

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.


Keywords


One-sided approximation; Characteristic function; Spherical layer; Spherical cap; Algebraic polynomials

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References


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DOI: http://dx.doi.org/10.15826/umj.2018.2.003

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