Spectral Theory of Orthogonal Polynomials

During the past dozen years, a major focus of my research has been the spectral theory of orthogonal polynomials—both orthogonal polynomials on the real line (OPRL) and on the unit circle (OPUC). There has been a flowering of the subject in part because of a cross-fertilization of two communities of researchers. I will discuss some aspects of this subject here; for a lot more, see my recent books on the subject. We begin with OPRL. If μ is a measure on C with ∫ |z| dμ < ∞ and so that μ is not supported on finitely many points, then {z}n=1 are independent in L (C, dμ) so one can use Gram–Schmidt to obtain monic and also normalized orthogonal polynomials. From the point of view of spectral/operator theory, two cases—OPRL (orthogonal polynomials on the real line) and OPUC (orthogonal polynomials on the unit circle)—are special because they have three-term recurrence relations which make the connection