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Rodrigues Formula for Jacobi Polynomials on the Unit Circle RODRIGUES FORMULA FOR JACOBI POLYNOMIALS ON THE UNIT CIRCLE MASTERS THESIS Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of the Ohio State University By Griffin Alexander Reiner-Roth Graduate Program in Mathematics The Ohio State University 2013 Master's Examination Committe: Professor Rodica Costin, Advisor Professor Barbara Keyfitz c Copyright by Griffin Alexander Reiner-Roth 2013 ABSTRACT We begin by discussing properties of orthogonal polynomials on a Borel measur- able subset of C. Then we focus on Jacobi polynomials and give a formula (analogous to that of [5]) for finding Jacobi polynomials on the unit circle. Finally, we consider some examples of Jacobi polynomials and demonstrate different methods of discov- ering them. ii ACKNOWLEDGMENTS I would like to thank my advisor, Dr. Rodica Costin, for all her help on this thesis; for finding a really interesting and doable project for me, for spending so much time explaining the basics of orthogonal polynomials to me, and for perfecting this paper. I also thank my parents for making sure I never starved. iii VITA 2011 . B.A. Mathematics, Vassar College 2011-Present . Graduate Teaching Associate, Department of Mathematics, The Ohio State Univer- sity FIELDS OF STUDY Major Field: Mathematics iv TABLE OF CONTENTS Abstract . ii Acknowledgments . iii Vita......................................... iv List of Tables . vii CHAPTER PAGE 1 Introduction . .1 1.1 Historical Background . .1 1.2 Orthogonality in Hilbert Spaces . .4 1.3 Orthogonal Polynomials with Respect to Measures . .5 1.4 Approximation by Orthogonal Polynomials . 12 1.5 Classical Orthogonal Polynomials . 15 2 Jacobi Polynomials . 17 2.1 Jacobi Polynomials on the Real Line . 17 2.2 Jacobi Polynomials on the Unit Circle . 19 2.3 Properties of Jacobi Polynomials on the Unit Circle . 21 2.4 Connection between Jacobi Polynomials on the Real Line and on the Unit Circle . 22 2.5 Calculating the Inner Product . 26 2.6 Formulas for Cn, Dn .......................... 32 2.7 Conjecture for An, Bn ......................... 35 3 New Results . 37 3.1 Commutation Relations . 37 3.2 Rodrigues' Formula . 39 v 4 Examples of Jacobi Polynomials on the Unit Circle . 41 4.1 Using the Gram-Schmidt Process . 41 4.2 Using Theorem 9 . 43 4.3 Using the New Rodrigues' Formula . 44 4.4 Comparison of Formulas . 45 5 Codes and Computer Calculation Results . 47 5.1 OPRL . 47 5.2 OPUC . 48 Bibliography . 57 vi LIST OF TABLES TABLE PAGE 5.1 Table 1: Comparison of Rodrigues' formulas on the real line ..... 49 5.2 Table 2: Computation times using the new Rodrigues' formula ... 54 vii CHAPTER 1 INTRODUCTION 1.1 Historical Background The study of orthogonal polynomials has a rich history, which can be traced to as early as the late eighteenth century, linked to Legendre's study of planetary motion. About a century later, orthogonal polynomials arose (seemingly, in a more natural manner) when searching for solutions to Sturm-Liouville problems. These polynomi- als are referred to as the classical orthogonal polynomials (we discuss properties of these polynomials in Section 1.5). Some of the first mathematicians to study these polynomials are Chebyshev, Markov, and Stieltjes. These three names come up again, in connection with orthogonal polynomials, regarding the Markov-Stieltjes inequality, which \plays a fundamental role in the theory of momentum probems" [7]. Cheby- shev conjectured the inequality, and Markov and Stieltjes (independently) proved the result. Orthogonal polynomials are defined on the real line (OPRL), or on the unit circle (OPUC); there is far more research that has been done on orthogonal polynomials on the real line. Simon [17] conjectures one reason is \[OPRL] examples appear in so many places that most scientists are exposed to them early in their education. 1 The applications of OPUC are subtle and beautiful, but less concrete." However, Szeg¨oproved a relation between two special classes of these sets of polynomials, so an additional application of OPUC is that knowledge of these polynomials can illuminate OPRL facts. In Section 2.4, we state and prove this relation. In the early twentieth century, Szeg¨o,the father of OPUC, is responsible for the revolution in the study of orthogonal polynomials by considering a wide variety of problems. In many ways, OPUC are analogous to OPRL. As an example, polynomials from both sets possess the Christoffel-Darboux Formula; for more information on the Christoffel-Darboux Formula in both settings, see Simon [17], Szeg¨o[19] (to whom the formula is due), and Freud [7] (whose work on the unit circle relies heavily on the minimum problem, i.e., minimizing the Christoffel function). In fact, \The study of the minimum problem is central to Szeg¨o'sinitial papers and have been a recurrent theme since" [17]. Other analogues on the unit circle include Favard's Theorem (a recent proof is given in [6]), Gauss-Jacobi quadrature, and the fact that they all satisfy three-term recursion relations. We conclude this section with a short summary of some areas of mathematics that have benefitted from the study of OPRL, and to a lesser extent, OPUC. We mentioned quadrature, an area that can be considered independently of orthogo- nal polynomials. Asymptotic behavior of orthogonal polynomials is a large field of study, and has applications to problems such as to Riemann-Hilbert problems. The asymptotic behavior of the ultraspherical and Laguerre polynomials (examples of clas- sical orthogonal polynomials) have been studied by Darboux and Fej´er,respectively. Asymptotic properties of orthogonal polynomials have continued to be investigated, 2 and results have been found by several mathematicans, from Szeg¨oto Simon. In- terpolation procedures, very important in applications, have been studied by, among others, Geronimus. A solution to the momentum problem (or the Hamburger momen- tum problem, named after Hamburger for his work on the problem) was proved due in no small part to results of orthogonal polynomials. Approximation theory (with profound results due to Nevai), Lagrange interpolation, continued fractions, stochas- tic processes, and even coding theory are subjects with results relying on orthogonal polynomials. 3 1.2 Orthogonality in Hilbert Spaces Orthogonal polynomials are the Rodney Dangerfield of analysis. Barry Simon, OPUC on One Foot Let (H; h·; ·i) be a Hilbert space and k · k be the associated norm, defined by kfk = phf; fi. Definition 1. Let F ⊆ H be a countable set of functions ffk : k ≥ 0g. fi is orthog- onal to fj if hfi; fji = 0, and F is an orthogonal set if for i 6= j, hfi; fji = 0. The family F is an orthonormal set if F is an orthogonal set and, in addition, kfik = 1 for each i ≥ 0. Let F be a countable, linearly independent set of functions. Then there exists an orthogonal set G in bijection with F such that span(F) = span(G). Such a G can be constructed using the Gram-Schmidt process: Theorem 2. (The Gram-Schmidt Process) Let n 2 N and ffk : 0 ≤ k ≤ ng be a linearly independent set of functions. Define g0 = f0 and for 1 ≤ k ≤ n, k−1 X hgj; fki g = f − g : k k kg k2 j j=0 j Then fgk : 0 ≤ k ≤ ng is an orthogonal set and spanffkg = spanfgkg. (For a proof of this theorem, see, for example, Sadun [15].) The process above can be extended to a countable family of linearly independent functions. We remark that one can obtain an orthonormal set from an orthogonal set by replacing each gk g with k . kgkk 4 1.3 Orthogonal Polynomials with Respect to Measures Let X be an interval in R or the unit circle and µ a positive Borel measure on X. Define Z 2 2 H = L (X; dµ) = f : X ! C: f is measurable and jfj dµ < 1 ; X the space of square integrable functions with respect to the measure µ. The binary operation defined by Z hf; gi = fg dµ (1.3.1) X for f, g 2 H is an inner product on H and it is well-known that (H; h·; ·i) becomes a Hilbert space. Notice that this inner product is finite by the Cauchy-Schwarz Inequality: Z Z jhf; gij = fg dµ ≤ jfj · jgj dµ X X Z 1=2 Z 1=2 ≤ jfj2 dµ jgj2 dµ < 1: X X Z Assumptions. We assume that the moments exist, that is, zk dµ is defined for X all k ≥ 0 (later, we give a more general definition of moments, but this definition will Z be sufficient on the unit circle). Further, assume µ is normalized so that 1 dµ = 1. X The set of polynomials fzk : k ≥ 0g is linearly independent, so one may apply Gram-Schmidt to this set to obtain the set of monic orthogonal polynomials of degree n ≥ 0. These are the orthogonal polynomials on X with respect to the measure µ. Throughout, the set f'ng = f'n : n ≥ 0g will denote the set of monic orthogonal polynomials, where 'n(z) is degree n (i.e., for each n 2 N, there exist complex num- n−1 n X k bers ak such that 'n(z) = z + akz ). In certain situations, the set of orthonormal k=0 5 polynomials is desired; denote this set by fΦng. In more theoretical settings, f'ng is usually of interest simply because monic polynomials are easier to work with. In the context of applications, fΦng may be preferred for its use in approximation. The first apparent difference between 'n and Φn involves the leading coefficient of Φn, which we denote by κn: n Φn(z) = κnz + ··· : 'n −1 Of course, Φn = , therefore, κn = k'nk .
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