The best extension of algebraic polynomials from the unit circle

A. V. Parfenenkov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 15 (2009) no. 1, pp. 184-194

Résumé

We consider the class $\mathfrak P_n$ of algebraic polynomials $P_n(x,y)$ of two variables of degree $n$ whose uniform norm on the unit circle $\Gamma_1$ centered at the origin is at most 1: $\|P_n\|_{C(\Gamma_1)}\le1$. We study the extension of polynomials from the class $\mathfrak P_n$ to the plane with the least uniform norm on the concentric circle $\Gamma_r$ of radius $r$. We prove that the values $\theta_n(r)$ of the best extension of the class $\mathfrak P_n$ satisfy the equalities $\theta_n(r)=r^n$ for $r>1$ and $\theta_n(r)=r^n-1$ for $0$.

Détail

Keywords: polynomial of many variables, the best extension, uniform norm
Mots-clés : harmonic polynomial.