Chebyshev's recursion: Analytic principles and applications

Abstract

We study different algebraic and algorithmic constructions related to the scalar product on the space of polynomials defined on the real axis and on the unit circle and to the Chebyshev procedure. A modern version of the Chebyshev recursion ($(m) - T$-recursion) is applied to check whether the Hankel and Toeplitz quadratic forms are positive definite, to determine the number of real (complex conjugate) roots of the polynomials, to localize the ordering of these roots, and to find bounds for the values of a function on a given set. We also consider the relation between the $(m) - T$-recursion and the method of moments in the study of Schrödinger operators for special classes of potentials.