Continued fractions which correspond to two series expansions and the strong Hamburger moment problem

Just as the denominator polynomials of a J-fraction are orthogonal polynomials with respect to some moment functional, the denominator polynomials of an M-fraction are shown to satisfy a skew orthogonality relation with respect to a stronger moment functional. Many of the properties of the numerators and denominators of an M- fraction are also studied using this pseudo orthogonality relation of the denominator polynomials. Properties of the zeros of the denominator polynomials when the associated moment functional is positive definite are also considered. A type of continued fraction, referred to as a J-fraction, is shown to correspond to a power series about the origin and to another power series about infinity such that the successive convergents of this fraction include two more additional terms of anyone of the power series. Given the power series expansions, a method of obtaining such a J-fraction, whenever it exists, is also looked at. The first complete proof of the so called strong Hamburger moment problem using a continued fraction is given. In this case the continued fraction is a J-fraction. Finally a special class of J-fraction, referred to as positive definite J-fractions, is studied in detail. The four chapters of this thesis are divided into sections. Each section is given a section number which is made up of the chapter number followed by the number of the section within the chapter. The equations in the thesis have an equation number consisting of the section number followed by the number of the equation within that section. In Chapter One, in addition to looking at some of the historical and recent developments of corresponding continued fractions and their applications, we also present some preliminaries. Chapter Two deals with a different approach of understanding the properties of the numerators and denominators of corresponding (two point) rational functions and, continued fractions. This approach, which is based on a pseudo orthogonality relation of the denominator polynomials of the corresponding rational functions, provides an insight into understanding the moment problems. In particular, results are established which suggest a possible type of continued fraction for solving the strong Hamburger moment problem. In the third chapter we study in detail the existence conditions and corresponding properties of this new type of continued fraction, which we call J-fractions. A method of derivation of one of these 3-fractions is also considered. In the same chapter we also look at the all important application of solving the strong Hamburger moment problem, using these 3-fractions. The fourth and final chapter is devoted entirely to the study of the convergence behaviour of a certain class of J-fractions, namely positive definite J-fractions. This study also provides some interesting convergence criteria for a real and regular 3-fraction. Finally a word concerning the literature on continued fractions and moment problems. The more recent and up-to-date exposition on the analytic theory of continued fractions and their applications is the text of Jones and Thron [1980]. The two volumes of Baker and Graves-Morris [1981] provide a very good treatment on one of the computational aspects of the continued fractions, namely Pade approximants. There are also the earlier texts of Wall [1948] and Khovanskii [1963], in which the former gives an extensive insight into the analytic theory of continued fractions while the latter, being simpler, remains the ideal book for the beginner. In his treatise on Applied and Computational Complex Analysis, Henrici [1977] has also included an excellent chapter on continued fractions. Wall [1948] also includes a few chapters on moment problems and related areas. A much wider treatment of the classical moment problems is provided in the excellent texts of Shohat and Tamarkin [1943] and Akhieser [1965]

Sieved para-orthogonal polynomials on the unit circle

We look at the para-orthogonal polynomials, chain sequences and quadrature formulas that follow from the kernel polynomials of sieved orthogonal polynomials on the unit circle

Orthogonal Polynomials Associated with Complementary Chain Sequences

Using the minimal parameter sequence of a given chain sequence, we introduce the concept of complementary chain sequences, which we view as perturbations of chain sequences. Using the relation between these complementary chain sequences and the corresponding Verblunsky coefficients, the para-orthogonal polynomials and the associated Szegő polynomials are analyzed. Two illustrations, one involving Gaussian hypergeometric functions and the other involving Carathéodory functions are also provided. A connection between these two illustrations by means of complementary chain sequences is also observed

Direct and inverse spectral transform for the relativistic Toda lattice and the connection with Laurent orthogonal polynomials

We introduce a spectral transform for the finite relativistic Toda lattice (RTL) in generalized form. In the nonrelativistic case, Moser constructed a spectral transform from the spectral theory of symmetric Jacobi matrices. Here we use a non-symmetric generalized eigenvalue problem for a pair of bidiagonal matrices (L,M) to define the spectral transform for the RTL. The inverse spectral transform is described in terms of a terminating T-fraction. The generalized eigenvalues are constants of motion and the auxiliary spectral data have explicit time evolution. Using the connection with the theory of Laurent orthogonal polynomials, we study the long-time behaviour of the RTL. As in the case of the Toda lattice the matrix entries have asymptotic limits. We show that L tends to an upper Hessenberg matrix with the generalized eigenvalues sorted on the diagonal, while M tends to the identity matrix.Comment: 24 pages, 9 figure