Complex Jacobi matrices and quadrature rules

Abstract

Given any sequence of orthogonal polynomials, satisfying the three term recurrence relation xpn(x) = ?n+1pn+1(x) + ?npn(x) + ?npn-1(x), p-1(x)=0 po(x)=1 with ?n ? 0, n ? N, ?0 = 1, an infinite Jacobi matrix can be associated in the following way ? ? ??0 ?1 0...? ? ? ??1 ?1 ?2...? ? ? J = ?0 ?2 ?2...? ?.... ? ?.... ? ?....? ? ? In the general case if the sequences {?n} or {?n} are complex the associated Jacobi matrix is complex. Under the condition that both sequences {?n}and {?n} are uniformly bounded, the associated Jacobi matrix can be understood as a linear operator J acting on ?2, the space of all complex square-summable sequences, where the value of the operator J at the vector x is a product of an infinite vector x and an infinite matrix J in the matrix sense. The case when the sequences {?n} and {?n} are not uniformly bounded, an operator acting on ?2 can not be defined that easily. Additional properties of the sequence of orthogonal polynomials are needed in order to be able to define the operator uniquely. The case when the sequences ?n and ?n are real is very well understood. The spectra of the Jacobi matrix J equals the support of the measure of orthogonality for the given sequence of orthogonal polynomials. All zeros of orthogonal polynomials are real, simple and interlace, contained in the convex hull of the spectra of the Jacobi operator associated with the infinite Jacobi matrix J. Every point in ?(J) attracts zeros of orthogonal polynomials. An application of orthogonal polynomials is the construction of quadrature rules for the approximation of integration with respect to the measure of orthogonality. For arbitrary sequences {?n} and {?n} the situation is changed dramatically. Zeros of orthogonal polynomials need not be simple; they are not real and they do not necessarily lie in the convex hull of ?(J). There is also a little known about convergence results of related quadrature rule. Only in recent years a connection between complex Jacobi matrices and related orthogonal polynomials is interesting again (see [2]). Studies of complex Jacobi operators should lead to a better understanding of related orthogonal polynomials, but also the study of orthogonal polynomials with the complex Jacobi matrices should put more light on the non-hermitian banded symmetric matrices. In this lecture some results are given about complex Jacobi matrices and related quadrature rules, and also some interesting examples are presented.