Best $L^2$-Extension of Algebraic Polynomials from the Unit Euclidean Sphere to a Concentric Sphere

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 $0<r<1$. We describe the best extension method. A.V. Parfenenkov obtained in 2009 a similar result in the uniform norm on the plane ($m=2$).