Рекурсия П. Л. Чебышева: некоторые аналитические и вычислительные аспекты

Abstract

We study different algebraic and algorithmic constructions related to both an inner product on the space of polynomials defined on the real axis and the unit circle, and the Chebyshev procedure. The modern variant of the Chebyshev recursion ((m)−T-recursion) is applied to check whether Hankel and Toeplitz quadratic forms are positive definite, to determine the number of real (complex conjugate) roots of a polynomial and to localize them, to find bounds on values of a function on a given set. We also consider the relation between (m)−T-recun>ion and the method of moments in the study of Schrodinger operator with the potential of a special class.