Sum Rules and the Szegő Condition for Orthogonal Polynomials on the Real Line
نویسنده
چکیده
We study the Case sum rules, especially C0, for general Jacobi matrices. We establish situations where the sum rule is valid. Applications include an extension of Shohat’s theorem to cases with an infinite point spectrum and a proof that if lim n(an− 1) = α and lim nbn = β exist and 2α < |β|, then the Szegő condition fails.
منابع مشابه
Bernstein–Szegő polynomials on the triangle
In this work we consider the extension of the one variable Bernstein–Szegő theory for orthogonal polynomials on the real line (see [3]) to bivariate measures supported on the triangle. A similar problem for measures supported in the square was studied in [1]. Following essentially [2] the orthogonal polynomials and the corresponding kernel functions are constructed. Finally, some asymptotic res...
متن کاملOn a Two Variable Class of Bernstein-szegő Measures
The one variable Bernstein-Szegő theory for orthogonal polynomials on the real line is extended to a class of two variable measures. The polynomials orthonormal in the total degree ordering and the lexicographical ordering are constructed and their recurrence coefficients discussed.
متن کاملJost functions and Jost solutions for Jacobi matrices, I. A necessary and sufficient condition for Szegő asymptotics
We provide necessary and sufficient conditions for a Jacobi matrix to produce orthogonal polynomials with Szegő asymptotics off the real axis. A key idea is to prove the equivalence of Szegő asymptotics and of Jost asymptotics for the Weyl solution. We also prove L2 convergence of Szegő asymptotics on the spectrum.
متن کاملORTHOGONAL ZERO INTERPOLANTS AND APPLICATIONS
Orthogonal zero interpolants (OZI) are polynomials which interpolate the “zero-function” at a finite number of pre-assigned nodes and satisfy orthogonality condition. OZI’s can be constructed by the 3-term recurrence relation. These interpolants are found useful in the solution of constrained approximation problems and in the structure of Gauss-type quadrature rules. We present some theoretical...
متن کاملSome new families of definite polynomials and the composition conjectures
The planar polynomial vector fields with a center at the origin can be written as an scalar differential equation, for example Abel equation. If the coefficients of an Abel equation satisfy the composition condition, then the Abel equation has a center at the origin. Also the composition condition is sufficient for vanishing the first order moments of the coefficients. The composition conjectur...
متن کامل