Abstract
We consider the problem of extending algebraic polynomials from the unit sphere of the Euclidean space of dimension \(m\geq 2\) to a concentric sphere of radius \(r\neq 1\) with the smallest value of the \(L^{2}\)-norm. An extension of an arbitrary polynomial is found. As a result, we obtain the best extension of a class of polynomials of given degree \(n\geq 1\) whose norms in the space \(L^{2}\) on the unit sphere do not exceed 1. We show that the best extension equals \(r^{n}\) for \(r>1\) and \(r^{n-1}\) for \(0<r<1\). We describe the best extension method. A.V. Parfenenkov obtained in 2009 a similar result in the uniform norm on the plane (\(m=2\)).
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Funding
This work was supported by the Russian Academic Excellence Project (agreement between the Ministry of Education and Science of the Russian Federation and Ural Federal University no. 02.A03.21.0006 of August 27, 2013). The work of the first author was also supported by the Russian Foundation for Basic Research (project no. 18-01-00336).
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Translated from Trudy Instituta Matematiki i Mekhaniki UrO RAN, Vol. 26, No. 2, pp. 47 - 55, 2020 https://doi.org/10.21538/0134-4889-2020-26-2-47-55.
Translated by M. Deikalova
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Arestov, V.V., Seleznev, A.A. Best \(L^{2}\)-Extension of Algebraic Polynomials from the Unit Euclidean Sphere to a Concentric Sphere. Proc. Steklov Inst. Math. 313 (Suppl 1), S6âS13 (2021). https://doi.org/10.1134/S0081543821030020
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DOI: https://doi.org/10.1134/S0081543821030020