V.V. Arestov, A.A. Seleznev. Best $L^2$-extension of algebraic polynomials from the unit Euclidean sphere to a concentric sphere ... P. 47-55

We consider the problem of extending algebraic polynomials from the unit sphere of a Euclidean space of dimension $m\ge 2$ to a concentric sphere of radius $r\ne1$ 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 $n\ge 1$ 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 ($m=2$).

Keywords: polynomial, Euclidean sphere, $L^2$-norm, best extension.

Received January 10, 2020

Revised February 10, 2020

Accepted February 17, 2020

Funding Agency: This work was supported by the Russian Academic Excellence Project (agreement no. 02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University). The research of the first author was also supported by the Russian Foundation for Basic Research (project no. 18-01-00336).

Vitalii Vladimirovich Arestov, Dr. Phys.-Math. Sci., Ural Federal University, Yekaterinburg, 620083 Russia; Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108  Russia, e-mail: vitalii.arestov@urfu.ru 

Anton Aleksandrovich Seleznev, Ural Federal University, Yekaterinburg, 620083 Russia, e-mail: misterion3000@gmail.com

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Cite this article as: V.V.Arestov, A.A.Seleznev. Best L2-extension of algebraic polynomials from the unit Euclidean sphere to a concentric sphere. Trudy Instituta Matematiki i Mekhaniki URO RAN, 2020, vol. 26, no. 2, pp. 47–55.