Doctor of Philosophy (PhD)
To study the orthogonal polynomials, Asai, Kubo and Kuo recently have developed the multiplicative renormalization method. Motivated by infinite dimensional white noise analysis, it is an alternative to the computational part of the classical Gram-Schmidt process to find the orthogonal polynomials for a given measure. Instead of finding the orthogonal polynomials recursively as described in the Gram-Schmidt process, one analyzes different types of generating functions systematically in order to obtain polynomials after power series expansion. This work also produces the Jacobi-Szego parameters easily and paves the way for the study of one-mode interacting Fock spaces related to these parameters. They have verified the classical measures and their corresponding orthogonal polynomials. In this thesis, we take this to the next level in order to classify measures with certain generating functions which leads to some new measures. We will begin with a description of the mathematical background of the method and we will re-derive all measures with generating functions of the exponential type as did Meixner and Morris in different settings. Next we shall derive all measures with generating functions of the fractional type, which will yield new measures We will also show the relation between consecutive Gegenbauer measures and neigbouring Jacobi polynomials. Finally we will demonstrate the uniqueness of the uniform measure with a certain type of generating function
Document Availability at the Time of Submission
Release the entire work immediately for access worldwide.
Namli, Suat, "Multiplicative renormalization method for orthogonal polynomials" (2007). LSU Doctoral Dissertations. 1935.