Chebyshev’s Problem of the Moments of Nonnegative Polynomials

Abstract

We study the problem of P. L. Chebyshev (proposed in 1883) concerning the extreme values of moments of nonnegative polynomials with weight on the interval $[-1,1]$ at a fixed zero moment, as well as this problem in a more general form. In the case of the first moment , the problem was solved by P. L. Chebyshev (1883) in the case of unit weight and by G. Szegö (1927) for an arbitrary weight. We have previously obtained a solution to Chebyshev’s problem for moments of odd order, which is largely based on the monotonicity of the function $x^{2k+1}$, $k\in\mathbb{N}$. The function $x^{2k}$ is not monotone on the interval $[-1,1]$, and the problem for moments of even order becomes more difficult. The paper provides a solution to Chebyshev’s problem on the largest values of moments of even order for polynomials of even degree. The problem of the smallest value of the second moment for polynomials of even degree is solved under an additional condition for the weight.