Linear functionals on space of polynomials and their application to semiclassical orthogonal polynomials

Abstract

In a study of orthogonal polynomials, the orthogonality relation of orthogonal polynominals given by the intergal with respect to a weight function can de extended to the formal orthognality relation with respect to a linear functional on space of polynominals(we call it a moment functional) and it has many advantages. For example, we need not specify the orthogonality interval and can replace the recurrence relations expressed in terms of moments by the simple relations foar orthogonalizing functionals through the formal operations of the derivative or the multiplication by polynomials on linear functionals. In Chapter 1, we provided a topological structure on the space of polynomial so that a moment functional might act continuously on it and found a topological function space containing polynomials on which the formal $\delta$-series expansion of a moment functional could act continuously. In Chapter 2, we studied the concept of quasi-orthogonality and semiclassical orthogonal polynomials and used moment functionals to obtain several characterizations for semiclassical orthogonal polynomials. In particular, we showed that if orthogonal polynomials ${P_n(x)}_0^\infty$ satisfy a differential equation of the from (we call it a Bochner-Krall differential equation of order 2r) {\boldmath$\Sigma_{i=0}^{2r}$} $l_i(x)y(i)(x) = \lambda_ny(x)$ where all the $l_i(x)``$s are polynomials of degree less then or equal to i and do not depend on n, then they are always semiclassical. In Chapter 3, we used moment functionals to unify the proof of characterization theorems for classical orthogonal polynomials which are discussed separately and obtain a characterization for orthogonal polynomials satisfying a Bochner-Krall differential equation of order 4. This an extension of Hahn``s Theorem which states that if both ${P_n(x)_0^\infty}$ and ${P_n^1(x)}_1^\infty$ are orthogonal polynomials, they must be classical orthogonal polynomials. In Chapter 4, we considered Koornwinder``...