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Hilbert analysis for orthogonal polynomials
 Orthogonal Polynomials and Special Functions (E. Koelink and W. Van Assche eds.) Lecture Notes in Mathematics 1817 (2003
"... Summary. This is an introduction to the asymptotic analysis of orthogonal polynomials based on the steepest descent method for RiemannHilbert problems of Deift and Zhou. We consider in detail the polynomials that are orthogonal with respect to the modified Jacobi weight (1 − x) α (1 + x) β h(x) on ..."
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Summary. This is an introduction to the asymptotic analysis of orthogonal polynomials based on the steepest descent method for RiemannHilbert problems of Deift and Zhou. We consider in detail the polynomials that are orthogonal with respect to the modified Jacobi weight (1 − x) α (1 + x) β h(x) on [−1, 1] where α, β> −1 and h is real analytic and positive on [−1, 1]. These notes are based on joint work with
Orthogonal polynomials, measures and recurrences on the unit circle
 Trans. Amer. Math. Soc
, 1987
"... Abstract. New characterizations are given for orthogonal polynomials on the unit circle and the associated measures in terms of the reflection coefficients in the recurrence equation satisfied by the polynomials. 1. Introduction. Let ..."
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Cited by 9 (1 self)
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Abstract. New characterizations are given for orthogonal polynomials on the unit circle and the associated measures in terms of the reflection coefficients in the recurrence equation satisfied by the polynomials. 1. Introduction. Let
WKB and Turning Point Theory for Second Order Difference Equations: External Fields and Strong Asymptotics for Orthogonal Polynomials
, 905
"... Abstract. A LGWKB and Turning point theory is developed for three term recurrence formulas associated with monotonic recurrence coefficients. This is used to find strong asymptotics for certain classical orthogonal polynomials including Wilson polynomials. ..."
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Cited by 2 (0 self)
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Abstract. A LGWKB and Turning point theory is developed for three term recurrence formulas associated with monotonic recurrence coefficients. This is used to find strong asymptotics for certain classical orthogonal polynomials including Wilson polynomials.
A Simple Proof Of "Favard's Theorem" On The Unit Circle
"... . A very short constructive proof is given for the unit circle analogue of the "Favard Theorem" on the orthogonality of a system of polynomials satisfying a Szego type recurrence relation. In what follows we will adopt the following notation. D is the open unit disk, that is D = {z # C C ..."
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. A very short constructive proof is given for the unit circle analogue of the "Favard Theorem" on the orthogonality of a system of polynomials satisfying a Szego type recurrence relation. In what follows we will adopt the following notation. D is the open unit disk, that is D = {z # C C C : z < 1}, T = #D is the unit circle. For a given polynomial # k of degree k its reverse # # k is defined by # # k (z)=z k # k (1/z). Let {# n (d) # n=0 } be the orthonormal polynomials corresponding to a given finite positive Borel measure on T with infinite support, that is # n (d, z)=# n (d)z n + +# n( d, 0),# n >0, (1) and 1 2# # T # n (d, e i# )#m (d, e i# )d(#)=# nm,n ,m#0. (2) Let # n (d)=# n (d)/# n (d) denote the monic orthogonal polynomials. Then they satisfy the the Szego recursion # n (d, z)=z# n1 (d, z)+# n (d, 0)# # n1 (d, z),n=1,2,..., (3) This material is based upon work supported by the National Science Foundation under Grant Nos. DMS8814488 (first three auth...
LaguerreHahn Orthogonal Polynomials with respect to the Hahn Operator: Fourthorder Difference Equation for the rth Associated and the LaguerreFreud Equations for the Recurrence Coefficients 1
"... A ma famille, mes amis et à tous ceux qui croient à l’effort et œuvrent pour la justice, la paix et la dignité humaine. REMERCIEMENTS Tout d’abord, je remercie le Professeur Augustin BANYAGA qui me fait un grand honneur en acceptant de présider le jury de cette thèse. Je remercie également les Profe ..."
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A ma famille, mes amis et à tous ceux qui croient à l’effort et œuvrent pour la justice, la paix et la dignité humaine. REMERCIEMENTS Tout d’abord, je remercie le Professeur Augustin BANYAGA qui me fait un grand honneur en acceptant de présider le jury de cette thèse. Je remercie également les Professeurs Saïd BELMEHDI, JeanPierre EZIN, Wolfram KOEPF et Côme GOUDJO pour avoir accepté de faire partie du jury. J’exprime ici ma reconnaissance aux professeurs M. Norbert HOUNKONNOU et André RONVEAUX pour les efforts fournis et les sacrifices énormes consentis pour la codirection de cette thèse. Mes remerciements vont aussi à l’endroit du Professeur JeanPierre Ezin, Directeur de l’IMSP, pour sa constante sollicitude. Je remercie profondément le Service Allemand d’Echanges Universitaires (DAAD) qui, en m’octroyant une bourse doctorale, a rendu possible la réalisation de ce travail. Mes remerciements vont aussi à l’endroit de l’Université Nationale du Bénin et en particulier de l’Institut de Mathématiques et de Sciences Physiques pour l’hospitalité, le soutien financier et les sacrifices consentis tout au long de ma formation. Le séjour en Europe, de septembre 1997 à mars 1998, a été déterminant pour la finalisation de ce
Point Derivations on the l 1Algebra of Polynomial
"... Polynomial hypergroups are a very interesting class of hypergroups with a great variety of examples which are quite different from groups. So the L 1algebras of hypergroups have properties very distinguished to the L 1algebras of groups, in particular in the context of amenability and related cond ..."
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Polynomial hypergroups are a very interesting class of hypergroups with a great variety of examples which are quite different from groups. So the L 1algebras of hypergroups have properties very distinguished to the L 1algebras of groups, in particular in the context of amenability and related conditions. Being amenable the L 1algebra
Müntz Systems and MüntzLegendre Polynomials
, 1999
"... The MuntzLegendre polynomials arise by orthogonalizing the Muntz system ,...} with respect to the Lebesgue measure on [0, 1]. In this paper, di#erential and integral recurrence formulas for the MuntzLegendre polynomials are obtained. Interlacing and lexicographical properties of their zeros ..."
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The MuntzLegendre polynomials arise by orthogonalizing the Muntz system ,...} with respect to the Lebesgue measure on [0, 1]. In this paper, di#erential and integral recurrence formulas for the MuntzLegendre polynomials are obtained. Interlacing and lexicographical properties of their zeros are studied, and the smallest and largest zeros are universally estimated via the zeros of Laguerre polynomials. The uniform convergence of the Christo#el functions is proved equivalent to the nondenseness of the Muntz system, which implies that in this case the MuntzLegendre polynomials tend to 0 uniformly on closed subintervals of [0, 1). Some inequalities for Muntz polynomials are also investigated, in particular, a sharp Markov inequality is proved. 1.