Generalized extremal Yudin problems for polynomials

Two extremal problems of V.A. Yudin for polynomials in a more general setting are studied. In the first problem, among polynomials with nonnegative expansion coefficients in orthogonal polynomials on a segment $[-1,1]$, for which several successive moments and derivatives at the point $-1$ are equal to zero, a polynomial with a maximum non-negativity segment is searched. The cases of the solving of the problem are described in terms of the Krein property. In the second problem, among polynomials with zero boundary conditions and zero first two moments on the segment $[-1,1]$, a polynomial with a minimum segment symmetric about zero on which it is nonnegative and nonpositive outside is searched. For the second problem, a complete solution was obtained.