Orthogonal Polynomials from Jacobi to Simon
نویسندگان
چکیده
4 Where do orthogonal polynomials come from? 10 Continued fractions . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Padé approximation and rational interpolation . . . . . . . . . . 11 Moment problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Jacobi matrices and spectral theory of self-adjoint operators . . . 13 Quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Random matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
منابع مشابه
The Analytic Theory of Matrix Orthogonal Polynomials
We survey the analytic theory of matrix orthogonal polynomials. MSC: 42C05, 47B36, 30C10 keywords: orthogonal polynomials, matrix-valued measures, block Jacobi matrices, block CMV matrices
متن کاملFine Structure of the Zeros of Orthogonal Polynomials, Iv. a Priori Bounds and Clock Behavior Yoram Last and Barry Simon
We prove locally uniform spacing for the zeros of orthogonal polynomials on the real line under weak conditions (Jacobi parameters approach the free ones and are of bounded variation). We prove that for ergodic discrete Schrödinger operators, Poisson behavior implies positive Lyapunov exponent. Both results depend on a priori bounds on eigenvalue spacings for which we provide several proofs.
متن کاملFine Structure of the Zeros of Orthogonal Polynomials, Iv. a Priori Bounds and Clock Behavior
We prove locally uniform spacing for the zeros of orthogonal polynomials on the real line under weak conditions (Jacobi parameters approach the free ones and are of bounded variation). We prove that for ergodic discrete Schrödinger operators, Poisson behavior implies positive Lyapunov exponent. Both results depend on a priori bounds on eigenvalue spacings for which we provide several proofs.
متن کاملUpward Extension of the Jacobi Matrix for Orthogonal Polynomials
Orthogonal polynomials on the real line always satisfy a three-term recurrence relation. The recurrence coefficients determine a tridiagonal semi-infinite matrix (Jacobi matrix) which uniquely characterizes the orthogonal polynomials. We investigate new orthogonal polynomials by adding to the Jacobi matrix r new rows and columns, so that the original Jacobi matrix is shifted downward. The r new...
متن کاملA class of matrix-valued polynomials generalizing Jacobi polynomials
A hierarchy of matrix-valued polynomials which generalize the Jacobi polynomials is found. Defined by a Rodrigues formula, they are also products of a sequence of differential operators. Each class of polynomials is complete, satisfies a two-step recurrence relation, integral inter-relations, and quasi-orthogonality relations. 1. Motivation The understanding of matrix-valued orthogonal polynomi...
متن کامل