The Rate of the Smallest Value of the Weighted Measure of the Nonnegativity Set for Polynomials with Zero Mean Value on a Closed Interval

Let P-n (alpha) 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 integral(1)(-1) phi((alpha)) (t)p(n) (t)dx = 0. We study the problem on the smallest value mu(n) = inf{m(p(n)): p(n) is an element of p(n) (alpha)} of the weighted measure m (p(n)) = integral(phi(alpha))(chi(pn)) (t)dt of the set where p(n), is nonnegative. The x(P) order of mu(n), with respect to n is found: it is proved that mu(n) (alpha) asymptotic to n(-2(alpha+1)) as n -> infinity.