Geodesic
Best $L^2$-Extension of Algebraic Polynomials from the Unit Euclidean Sphere to a Concentric Sphere
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 26 (2020) no. 2, pp. 47-55

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We consider the problem of extending algebraic polynomials from the unit sphere of the 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 $0$. We describe the best extension method. A.V. Parfenenkov obtained in 2009 a similar result in the uniform norm on the plane ($m=2$).
Mots-clés : polynomial, $L^2$-norm
Keywords: Euclidean sphere, best extension. 
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     author = {V. V. Arestov and A. A. Seleznev},
     title = {Best $L^2${-Extension} of {Algebraic} {Polynomials} from the {Unit} {Euclidean} {Sphere} to a {Concentric} {Sphere}},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {47--55},
     publisher = {mathdoc},
     volume = {26},
     number = {2},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2020_26_2_a2/}
}
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V. V. Arestov; A. A. Seleznev. Best $L^2$-Extension of Algebraic Polynomials from the Unit Euclidean Sphere to a Concentric Sphere. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 26 (2020) no. 2, pp. 47-55. http://geodesic.mathdoc.fr/item/TIMM_2020_26_2_a2/