The rate of the smallest value of the weighted measure of the nonnegativity set for polynomials with zero mean value on a closed interval

Abstract

Let P _n (α) be the set of algebraic polynomials p _n of order n with real coefficients and zero weighted mean value with ultraspherical weight $\phi ^{(\alpha )} (t) = (1 - t^2 )^\alpha$ on the interval $[ - 1,1]:\int_{ - 1}^1 {\phi ^{(\alpha )} (t)p_n (t)dx = 0}$. We study the problem on the smallest value µ _n = inf{m(p _n ): p _n ∈ P _n (α)} of the weighted measure $m(p_n ) = \int_{\chi (p_n )} {\phi ^{(\alpha )} (t)dt}$ of the set where p _n is nonnegative. The order of µ _n with respect to n is found: it is proved that $\mu _n (\alpha ) \asymp n^{ - 2(\alpha + 1)}$ as n→∞.