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UniversiV M ic rm lm s International 300 N. Zeeb Road Ann Arbor, Ml 48106 8510547 Bauldry, William Charles ORTHOGONAL POLYNOMIALS ASSOCIATED WITH EXPONENTIAL WEIGHTS The Ohio State University Ph.D. 1985 University Microfilms I ntern StiO neI 300 N. zeeb Roaa, Ann Arbor, Ml 48106 ORTHOGONAL POLYNOMIALS ASSOCIATED WITH EXPONENTIAL WEIGHTS DISSERTATION Presented In Partial Fulfillment o f the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By William Charles Bauldry, B.S., M.A., M.5. The Ohio State University, 1985 Reading Committee: Approved By Prof. Paul Neval Prof. Bogdan BaishanskI Prof. Ranko Bojanic Vs] ______ Prof. William Davis Adviser Department o f Mathematics ACKNOWLEDGEMENTS 1 would like to thank my adviser, Dr. Paul Nevai, fo r all the support, help, and wisdom he has imparted to me during my graduate training. It was his inspiration and guidance that made this paper possible. A vote of thanks also goes to Drs. Doron Lubinsky and Attila Mate fo r the time and knowledge they shared with me in developing this dissertation project. Last of all, 1 wish to thank my family, especially my wife, Sue, fo r their moral support, encouragement and love. VITA December 14, 1950 ..................... Born - Mt. Clemens, Michigan 1975 ............................................ B.S. in Mathematics, Central Michigan University, Mt. Pleasant, Michigan 1975-1977 .................................. Teaching Associate, Department of Mathematics, Central Michigan University, Mt. Pleasant, Michigan 1977 ............................................. M.A. in Mathematics, Central Michigan University, Mt. Pleasant, Michigan 1977-1980 .................................. Lecturer, Department of Mathematics, The Ohio State University, Mansfield Campus. Mansfield, Ohio 1980-1985 .................................. Teaching Associate, Department of Mathematics, The Ohio State University, Columbus, Ohio 1983 ............................................. M.S. in Mathematics, The Ohio State University, Columbus, Ohio PUBLICATIONS "Estimates of Christoff el Functions of Generalized Freud-type Weights." Journal of Approximation Theory (to appear). II) FIELDS OF STUDY Major Field: Mathematics Studies in Approximation Theory. Professors Paul Nevai and Ranko Bojanic. Studies in Analysis. Professors Bogdan Baishanski and Gerald Edgar. IV TABLE OF CONTENTS PAGE ACKNOWLEDGEMENTS........................................................................ ii VITA .................................................................................................. iii INTRODUCTION................................................................................... 1 CHAPTER 1: ESTIMATES OF CHRISTOFFEL FUNCTIONS OF GENERALIZED FREUD-TYPE WEIGHTS........................................ 5 1. Introduction ........................................................................... 5 2. Notation ................................................................................. 7 3. ThB Main Results..................................................................... 9 4. Proof of the "Infinite to Finite Range" Inequality ............. 12 5. Proofs o f the Upper and Lower Bounds o f the Christoff el Functions ............................................................................. 17 6. Connections to the Orthonormal Polynomials Pp(w^;x) . 24 CHAPTER II: ASYMPTOTICS FOR THE RECURSION COEFFICIENTS a^ ANDb^ ASSOCIATED WITH THE ORTHOGONAL POLYNOMIALS WITH THE WEIGHT w(x) = exp{-Q(x)}........................................... 30 1. Introduction ............................................................................. 30 2. Notation ................................................................................... 34 3. The Main Results....................................................................... 35 4. Proof of the Existence Theorem fo r Asym ptotics ............... 38 5. The Asymptotic Series of the Recursion C oefficients 50 CHAPTER III: ASYMPTOTICS FOR THE ORTHOGONAL POLYNOMIALS ASSOCIATED WITH w(x) = exp{-Q(x)} ...................................... 62 L Introduction .......................................................................... 62 2. Notation ................................................................................. 65 3. The Main Results .................................................................... 67 4. A Preliminary Estimate o f pp(w:x) ...................................... 70 5. The Asymptotics of P p (w :0 ) ................................................. 81 6. Asymptotics for P p , ( w ; x ) ....................................................... 91 CHAPTER IV: PLANCHEREL-ROTACH TYPE ASYMPTOTICS FOR THE ORTHOGONAL POLYNOMIALS ASSOCIATED WITH w(x) = exp{-Q(x)}........................................................................ 105 1. Introduction .......................................................................... 105 2. Notation ................................................................................. 108 3. The Main Result ....................................................................... 110 4. Lemmata ................................................................................. 112 5. Proof of The Plancherel-Rotach Type Asymptotics 126 BIBLIOGRAPHY..................................................................................... 140 VI INTRODUCTION "It is the essence o f Mathematics that it concerns itself with those relations which lie so deep in the nature of things that they recur in the most varied situations. ... Among these are the formulations relating to the general analytical concept of orthogonality ..." - D. Jackson, [Jal, p. v]. The notion o f perpendicular is fundamental to our perception o f the reality o f the space in which we live. This concept has its generalization in the abstract as orthogonality. Once we define a criterion o f measurement on a collection o f objects, we can create a definition o f orthogonal elements. To wit; let $ and f be two objects o f a universal set with a measurement denoted by ($,9) taking real values. $ and 9 will be called orthogonal if and only if ($,»)=0. When the universal set is the collection o f functions and the measurement is given by an inner product, we have orthogonal functions. If we further specify that the functions are polynomials and the inner product Is an integral, we have entered the realm of orthogonal polynomials. The study of orthogonal polynomials originated from the theory of continued fractions ( SzegojSzl] ). In its infancy, the theory was developed from the point o f view o f continued fractions and their relation to the moment problem iShoTal]; however, as the importance o f the theory was recognized,the starting point was shifted to the property of orthogonality in the following manner. Let ji(x) be a nondecreasing, bounded function on the real numbers taking real values. Define an inner product on the class L^(dp:E) by (*.9) =J $(x)Ÿ(x) djj(x), and we shall use our definition o f orthogonality given above. The Gram - Schmidt process of orthogonal izat ion (see e.g. Jackson, [Jal, p. 151]) can be applied to any collection of ji-integrable linearly independent functions to produce an orthogonal set. The set o f powers o f x { $j(x) = x' : 1er, r c N } ( r may be either o f finite or infinite cardinality ) gives rise, through this process, to the polynomials { pp(d;i; x ) } orthogonal with respect to dp(x). If we define a^ = where is the leading coefficient of Pp(d|i;x), then we immediately see that ap(d|i) = r X PpCdp; x)pp_,

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