Abstract
Semi-infinite matrices, generalized eigenvalue problems, and orthogonal polynomials are closely related subjects. They connect different domains in mathematics—matrix theory, operator theory, analysis, differential equations, etc. The classical examples are Jacobi and Hessenberg matrices, which lead to orthogonal polynomials on the real line (OPRL) and orthogonal polynomials on the unit circle (OPUC). Recently there turned out that pencils (i.e., operator polynomials) of semi-infinite matrices are related to various orthogonal systems of functions. Our aim here is to survey this increasing subject. We are mostly interested in pencils of symmetric semi-infinite matrices. The corresponding polynomials are defined as generalized eigenvectors of the pencil. These polynomials possess special orthogonality relations. They have physical and mathematical applications that will be discussed. Examples show that there is an unclarified relation to Sobolev orthogonal polynomials. This intriguing connection is a challenge for further investigations.
Highlights
We will introduce the main objects of this chapter along with some brief historical notes.By operator pencils or operator polynomials one means polynomials of a complex variable λ whose coefficients are linear bounded operators acting in a Banach space X: Xm LðλÞ 1⁄4 λ jA j, (1)j1⁄40 where A j : X ! X (j 1⁄4 0, ... , m), see, for example, [1, 2]
We shall be mainly interested in pencils of banded semi-infinite matrices that are related to different kinds of scalar orthogonal polynomials
The classical example of such a relation is the case of orthogonal polynomials on the real line
Summary
We will introduce the main objects of this chapter along with some brief historical notes. The classical example of such a relation is the case of orthogonal polynomials on the real line. The classical example is the case of orthogonal polynomials on the unit circle (OPUC) and the corresponding three-term recurrence relation, see ref. A natural generalization of OPRL is matrix orthogonal polynomials on the real line (MOPRL). It turned out that MOPRL is closely related to orthogonal polynomials on the radial rays in the complex plane, see refs. During last years there appeared several examples of Sobolev polynomials, which are eigenfunctions of pencils of differential or difference operators In ref. [12] there was studied a more general case of relation (4), with a polynomial hðλÞ instead of λN
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