On an extremal problem for polynomials in~$n$ variables

Yu. A. Brudnyi ; M. I. Ganzburg

Izvestiya. Mathematics , Tome 7 (1973) no. 2, pp. 345-356

Résumé

This article is devoted to an examination of the following extremal problem: find the quantity $$ C_{k,n}(\lambda,B)=\sup_{|\omega|\ge\lambda}\sup_{P\in\mathscr P_{k,n}(\omega)}\|P\|_{C(B)}, $$ where $B$ is an $n$-dimensional sphere and $\mathscr P_{k,n}(\omega)$ is the totality of polynomials $P$ of degree $k$ in $n$ variables for which $\|P\|_{C(\omega)}\le1$. Here $\omega$ is a measurable set from $B$ and the first sup is taken over all measurable $\omega\subset B$ having measure $|\omega|\ge\lambda$. The exact order of growth of $C_{k,n}(\lambda, B)$ which respect to $\lambda$ as $\lambda\to0$ is found in this article. Various applications of the results are examined as well.