C ONTEMPORARY M ATHEMATICS 507 (cid:28)(cid:8)(cid:26)(cid:8)(cid:13)(cid:25)(cid:4)(cid:29)(cid:22)(cid:8)(cid:13)(cid:30)(cid:9)(cid:4) (cid:12)(cid:13)(cid:4)(cid:31)(cid:22)(cid:25) (cid:23)(cid:7)(cid:23)(cid:13)(cid:6)(cid:20)(cid:4)!(cid:23)(cid:20)"(cid:13)(cid:23)#(cid:12)(cid:6)(cid:20)(cid:9)(cid:4) (cid:6)(cid:13)(cid:30)(cid:4)$(cid:5)(cid:5)(cid:22)(cid:23)%(cid:12)#(cid:6)(cid:25)(cid:12)(cid:23)(cid:13)(cid:4)(cid:29) (cid:8)(cid:23)(cid:22)" &(cid:13)(cid:25)(cid:8)(cid:22)(cid:13)(cid:6)(cid:25)(cid:12)(cid:23)(cid:13)(cid:6)(cid:20)(cid:4)’(cid:23)(cid:22)((cid:9) (cid:23)(cid:5)(cid:4)(cid:12)(cid:13)(cid:4))(cid:23)(cid:13)(cid:23)(cid:22)(cid:4)(cid:23)(cid:24)(cid:4) *+(cid:12)(cid:20)(cid:20)(cid:8)(cid:22)#(cid:23)(cid:4),-(cid:5)(cid:8).(cid:4),(cid:6)(cid:7)(cid:23)#(cid:6)(cid:9)(cid:12)(cid:13)(cid:23)/(cid:9)(cid:4)0(cid:19)(cid:25) (cid:4)1(cid:12)(cid:22)(cid:25) (cid:30)(cid:6)" (cid:11)(cid:8)(cid:5)(cid:25)(cid:8)#(cid:21)(cid:8)(cid:22)(cid:4)(cid:17)2(cid:2)(cid:3)3(cid:4)(cid:3)(cid:19)(cid:19)(cid:17) 4(cid:13)(cid:12)(cid:27)(cid:8)(cid:22)(cid:9)(cid:12)(cid:30)(cid:6)(cid:30)(cid:4)5(cid:6)(cid:22)(cid:20)(cid:23)(cid:9)(cid:4)&&&(cid:4)(cid:30)(cid:8)(cid:4)6(cid:6)(cid:30)(cid:22)(cid:12)(cid:30) ,(cid:8)(cid:7)(cid:6)(cid:13)7(cid:9)3(cid:4)(cid:11)(cid:5)(cid:6)(cid:12)(cid:13) 8(cid:23)(cid:22)(cid:7)(cid:8)(cid:4)$(cid:22)(cid:27)(cid:8)(cid:9)9 :(cid:22)(cid:6)(cid:13)(cid:26)(cid:12)(cid:9)(cid:26)(cid:23)(cid:4)6(cid:6)(cid:22)(cid:26)(cid:8)(cid:20)(cid:20);(cid:13) $(cid:13)(cid:30)(cid:22)(cid:8)(cid:12)(cid:4)6(cid:6)(cid:22)(cid:25)<(cid:13)(cid:8).=:(cid:12)(cid:13)((cid:8)(cid:20)(cid:9) (cid:25)(cid:8)(cid:12)(cid:13) >(cid:30)(cid:12)(cid:25)(cid:23)(cid:22)(cid:9) American Mathematical Society Recent Trends in Orthogonal Polynomials and Approximation Theory This page intentionally left blank C ONTEMPORARY M ATHEMATICS 507 Recent Trends in Orthogonal Polynomials and Approximation Theory International Workshop in Honor of Guillermo López Lagomasino's 60th Birthday September 8–12, 2008 Universidad Carlos III de Madrid Leganés, Spain Jorge Arvesú Francisco Marcellán Andrei Martínez-Finkelshtein Editors American Mathematical Society Providence, Rhode Island Editorial Board Dennis DeTurck, managing editor George Andrews Abel Klein Martin J. Strauss 2000 Mathematics Subject Classification. Primary 15A52, 26C10, 30E10,31A15, 31C15, 33C47, 41A20, 42C05, 44A60, 65D32. Library of Congress Cataloging-in-Publication Data International Workshop on Orthogonal Polynomials and Approximation Theory (2008: Univer- sidadCarlosIIIdeMadrid) Recent trends in orthogonal polynomials and approximation theory : conference in honor of Guillermo Lo´pez Lagomasino’s 60th birthday, September 8–12, 2008, Universidad Carlos III de Madrid, Legan´es, Spain / Jorge Arvesu´, Francisco Marcella´n, Andrei Mart´ınez-Finkelshtein, editors. p.cm. —(Contemporarymathematics;v.507) Includesbibliographicalreferences. ISBN978-0-8218-4803-6(alk.paper) 1. Orthogonal polynomials—Congresses. 2. Approximation theory—Congresses. I. Lo´pez Lagomasino,Guillermo,1948– II.Arvesu´,Jorge,1968– III.Marcell´an,Francisco. IV.Mart´ınez- Finkelshtein,Andrei. V.Title. QA404.5.I587 2010 515(cid:1).55—dc22 2009040384 Copying and reprinting. Materialinthisbookmaybereproducedbyanymeansfor edu- cationaland scientific purposes without fee or permissionwith the exception ofreproduction by servicesthatcollectfeesfordeliveryofdocumentsandprovidedthatthecustomaryacknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercialuseofmaterialshouldbeaddressedtotheAcquisitionsDepartment,AmericanMath- ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can [email protected]. Excludedfromtheseprovisionsismaterialinarticlesforwhichtheauthorholdscopyright. In suchcases,requestsforpermissiontouseorreprintshouldbeaddresseddirectlytotheauthor(s). (Copyrightownershipisindicatedinthenoticeinthelowerright-handcornerofthefirstpageof eacharticle.) (cid:1)c 2010bytheAmericanMathematicalSociety. Allrightsreserved. TheAmericanMathematicalSocietyretainsallrights exceptthosegrantedtotheUnitedStatesGovernment. Copyrightofindividualarticlesmayreverttothepublicdomain28years afterpublication. ContacttheAMSforcopyrightstatusofindividualarticles. PrintedintheUnitedStatesofAmerica. (cid:1)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageathttp://www.ams.org/ 10987654321 151413121110 With our best wishes to Guillermo L´opez Lagomasino on the occasion of his 60th birthday. This page intentionally left blank Contents Preface ix Guillermo L´opez Lagomasino: Mathematical life Francisco Marcella´n and Andrei Mart´ınez-Finkelshtein 1 A walk through approximation theory Bernardo de la Calle Ysern 25 Asymptotic uniqueness of best rational approximants to complex Cauchy transforms in L2 of the circle Laurent Baratchart and Maxim Yattselev 87 Quadrature rules on the unit circle. A survey. Luis Garza and Francisco Marcella´n 113 On the multilinear trigonometric problem of moments Alberto Ibort, Pablo Linares, and Jose G. LLavona 141 Multiple orthogonal polynomial ensembles Arno B. J. Kuijlaars 155 Some equivalent formulations of universality limits in the bulk Eli Levin and Doron S. Lubinsky 177 Greedy energy points with external fields Abey Lo´pez Garc´ıa 189 On asymptotic behavior of Heine-Stieltjes and Van Vleck polynomials Andrei Mart´ınez-Finkelshtein and Evgenii A. Rakhmanov 209 Remarks on relative asymptotics for general orthogonal polynomials Edward B. Saff 233 Fine structure of the zeros of orthogonal polynomials: A progress report Barry Simon 241 A potential-theoretic problem connected with complex orthogonality Herbert Stahl 255 Orthogonal polynomials and approximation theory: Some open problems Walter Van Assche 287 vii This page intentionally left blank Preface This volume contains articles by speakers in the International Workshop on OrthogonalPolynomialsandApproximationTheory,heldonSeptember8-12,2008, at the Legan´es campus of the Universidad Carlos III de Madrid, in honor of the 60th birthday of Professor Guillermo L´opez Lagomasino. The five-day conference attracted140participantsfrom22countries. Theprogramoftheconference,video, and pdf files of most of the talks can be found at http://gama.uc3m.es. This volume’s articles cover a wide range of topics in the theory of Orthogonal Polynomials as well asin RationalApproximation witha specialemphasis ontheir applications in random matrices, integrable systems, and numerical quadrature. The contributors are celebrated researchers in these domains. A brief account of the scientific contributions by Professor Guillermo L´opez Lagomasino constitutes one of the features of this volume. His activity in the for- merSoviet Union, inthe framework of the group lead by A. A.Gonchar, as well as theirconnectionswiththeWesterngroupsworkinginOrthogonalPolynomials and Approximation Theory, fulfils a good historical perspective of the internationaliza- tionofthesedomainsinthelast50years. Thisisanexampleofhowmathematicians areabletoopenphysicalandpoliticalbarriersandcontributetotheimprovedlinks between individuals and institutions. New results and methods are presented in the contributed papers as well as a carefulchoiceofopenproblemstofosteraninterestinresearchinthesemathemati- calareasinthecomingyears. Thereaderwillfindsurveypresentations,anaccount ofrecentdevelopments,andtheexpositionofnewtrendsintheareasofOrthogonal Polynomials,SpecialFunctions,andApproximationTheory,fromatheoreticaland an applied perspectives. Asco-organizersoftheworkshopandeditorsofthisvolumeitisourhappytask tothankthoseindividualsandinstitutionswhoseeffortsmadeitpossible. First,we acknowledge Ministerio de Educaci´on y Ciencia of Spain (grant MTM2007-31221- E), Proyecto Ingenio Mathematica (i-MATH)(grant C3-0116), Universidad Car- los III de Madrid, Universidad Polit´ecnica de Madrid, Real Sociedad Matem´atica Espan˜ola, Sociedad Espan˜ola de Matem´atica Aplicada, Ayuntamiento de Legan´es, and International Association for the promotion and cooperation with scientists (INTAS) for their financial support. Second, it is a pleasure to thank all the mem- bers of the local organization Committee of the Universidad Carlos III de Madrid for the excellent organization of this meeting. Last, but certainly not least, we ex- press our gratitude to the participants of the workshop who made it a memorable ix