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

Algebraic functions;

Sri Ranga, A. ; McCabe, J. H. ; McCabe, J. H.

University:University of St Andrews

Department:Mathematics & Statistics (School of)

关键词: Algebraic functions;   

Others  :  https://research-repository.st-andrews.ac.uk/bitstream/handle/10023/2977/ASriRangaPhDThesis.pdf?sequence=3&isAllowed=y

来源: DR-NTU

【 摘 要 】

Just as the denominator polynomials of a J-fraction areorthogonal polynomials with respect to some moment functional, thedenominator polynomials of an M-fraction are shown to satisfy a skeworthogonality 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 relationof the denominator polynomials. Properties of the zeros of thedenominator polynomials when the associated moment functional ispositive definite are also considered.A type of continued fraction, referred to as a J-fraction, isshown to correspond to a power series about the origin and to anotherpower series about infinity such that the successive convergents ofthis fraction include two more additional terms of anyone of thepower series. Given the power series expansions, a method ofobtaining such a J-fraction, whenever it exists, is also looked at.The first complete proof of the so called strong Hamburger momentproblem using a continued fraction is given. In this case thecontinued fraction is a J-fraction.Finally a special class of J-fraction, referred to as positivedefinite 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 thechapter number followed by the number of the section within thechapter. The equations in the thesis have an equation numberconsisting of the section number followed by the number of theequation within that section.In Chapter One, in addition to looking at some of thehistorical and recent developments of corresponding continuedfractions and their applications, we also present some preliminaries.Chapter Two deals with a different approach of understandingthe properties of the numerators and denominators of corresponding(two point) rational functions and, continued fractions. Thisapproach, which is based on a pseudo orthogonality relation of thedenominator polynomials of the corresponding rational functions,provides an insight into understanding the moment problems. Inparticular, results are established which suggest a possible typeof continued fraction for solving the strong Hamburger momentproblem.In the third chapter we study in detail the existenceconditions and corresponding properties of this new type of continuedfraction, which we call J-fractions. A method of derivation of oneof these 3-fractions is also considered. In the same chapter we alsolook at the all important application of solving the strong Hamburgermoment problem, using these 3-fractions.The fourth and final chapter is devoted entirely to the studyof the convergence behaviour of a certain class of J-fractions,namely positive definite J-fractions. This study also provides someinteresting convergence criteria for a real and regular 3-fraction.Finally a word concerning the literature on continued fractionsand moment problems. The more recent and up-to-date exposition onthe analytic theory of continued fractions and their applications isthe text of Jones and Thron [1980]. The two volumes of Baker andGraves-Morris [1981] provide a very good treatment on one of thecomputational aspects of the continued fractions, namely Padeapproximants. There are also the earlier texts of Wall [1948] andKhovanskii [1963], in which the former gives an extensive insightinto the analytic theory of continued fractions while the latter,being simpler, remains the ideal book for the beginner. In histreatise 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 andrelated areas. A much wider treatment of the classical momentproblems is provided in the excellent texts of Shohat and Tamarkin[1943] and Akhieser [1965].