Kernels on the unit circle. Orthogonality

Abstract

In the study of orthogonal polynomials on the unit circle T , one can only state the orthogonality of the kernel polynomials in very few situations (Tasis, 1989). Nevertheless, in the study of orthogonal polynomials on the unit circle one can pose the problem in the same terms as in the real line: Given a regular and hermitian functional u in the space of Laurent polynomials one can consider its modified form by a polynomial of degree one, ( z − α) u. Since this new functional is not hermitian for any complex α, we cannot consider orthogonality in the usual sense. Instead we speak about biorthogonality (Suárez, 1993) with respect to ( z − α) u. In the present paper we focus our attention on this last problem. In Section 1 we study the regularity of ( z − α) u under the assumption on the regularity of u. One of the main results is Theorem 1.3 in which we prove that the right monic orthogonal polynomial sequence related to ( z − α) u is the sequence of monic kernel polynomials K ̃ n(z,α) . In case of orthogonality in the usual sense we also obtain, in Theorems 1.8 and 1.9, the functionals and the measures of orthogonality in the positive definite case. In Section 2 we give the differential equation satisfied by the sequence K n ( z, α) nϵ N in the semiclassical case. Finally, in the last section, we study some properties concerning the roots of kernel polynomials and, in order to compute the roots, we obtain that the zeros of K n ( z, α) are the eigenvalues of a certain matrix.