Characterization of orthogonal polynomials with respect to a functional

Abstract

In this paper we study questions of existence, uniqueness and characterization of polynomials orthogonal with respect to a linear, not necessarily definite, functional L defined on the set of Laurent polynomials. First we characterize with the help of the function F L(z):= L( (y + z) (y − z) ) polynomials orthogonal with respect to L . Using this characterization, which has wide applications, we are able to settle the question of existence and uniqueness of orthogonal polynomials. The uniquely determined orthogonal polynomials will be called basic orthogonal polynomials. It is to be pointed out that, in contrast to the real case, there are natural numbers n μ such that there exist no polynomials of degree n μ which are orthogonal with respect to L , if L is indefinite and if we have “orthogonality-jumps” greater than 1. Furthermore the functional to which the basic orthogonal polynomials of the second kind are orthogonal is determined. Finally, we get explicit expressions for all basic orthogonal polynomials with respect to a “weight function” the support of which consists of several arcs of the unit circle, changes sign from arc to arc and has square root singularities at the boundary points of the arcs. These polynomials can be considered as the basic polynomials in describing and generating orthogonal polynomials with periodic reflection coefficients.