<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN"            "http://www.w3.org/TR/REC-html40/loose.dtd"> <html> <meta name="GENERATOR" content="TtH 3.11">  <style type="text/css"><!--  td div.comp { margin-top: -0.6ex; margin-bottom: -1ex;}  td div.comb { margin-top: -0.6ex; margin-bottom: -.6ex;}  td div.hrcomp { line-height: 0.9; margin-top: -0.8ex; margin-bottom: -1ex;}  td div.norm {line-height:normal;}  span.roman {font-family: serif; font-style: normal; font-weight: normal;}   span.overacc2 {position: relative;  left: .8em; top: -1.2ex;}  span.overacc1 {position: relative;  left: .6em; top: -1.2ex;} --></style>        <center>            <font size="+2"> <title>CURRICULUM VITAE</title>  CURRICULUM VITAE</font> </center>  <p> <br /><br /> Nom&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <font size="+2">Alphonse P. Magnus</font>  <p> <br /> Date et lieu de naissance &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 23 ao&#251;t 1946, Uccle.  <p> <br /> Adresse  <p> &nbsp;priv&#233;e:  <p>    &nbsp;Chauss&#233;e de Mont-Saint-Pont,17  <p> &nbsp;1440 Braine-le-Ch&#226;teau  <p> &nbsp;bureau:  <p>    &nbsp;Universit&#233;  Catholique de Louvain  <p> &nbsp;Institut Math&#233;matique  <p> &nbsp;Chemin du Cyclotron,2  <p> &nbsp;1348 Louvain-la-Neuve.  <p> &nbsp;Tel. 010 473157  <p> &nbsp;e-mail&nbsp;<tt>&nbsp;magnus@anma.ucl.ac.be</tt>  <p> <br /> Titres  <p>          Ing&#233;nieur civil en math&#233;matiques appliqu&#233;es (grade scientifique), UCL, 1969.  <p>  Docteur en sciences appliqu&#233;es (grade scientifique), UCL, 1976.  <p> <br /> Nominations  <p>   Assistant, UCL, 1969,  <p>  Premier assistant, UCL, 1976,  <p>  Chef de travaux, UCL, 1984.  <p>  Charg&#233; de cours &#224; temps partiel, UCL, 1997.  <p> <br /> Discipline  <p>   Math&#233;matiques, Analyse num&#233;rique, Th&#233;orie de l'approximation.  <p> <br /><font size="+2">Enseignement, encadrement:</font>  <p> Encadrement exercices analyse num&#233;rique en MATH21, MAP21, MAP22 et PHYS 21-22  (aussi INFO21 ou 22  1980-1990).  <p>   MATH2171, Analyse num&#233;rique 1a : depuis 1972  <p>   MATH2172, Analyse num&#233;rique 1b :  1972-1994  <p> Encadrement cours et exercices analyse FSA11 et 12: quelques int&#233;rims,  <p> Encadrement exercices programmation FSA11 et/ou MATH12 : p&#233;riode 1976-1980.  <p> Directions r&#233;guli&#232;res de m&#233;moires MAP23 et MATH22.  <p> <br />Cours  <p> 2<sup>&#232;<span class="roman">me</span></sup> cycle:  <p>   ``Approximation complexe quantitative'', cours de questions sp&#233;ciales    en math&#233;matique,  <p>      Facult&#233;s N.D. de la Paix, Namur, janvier-mai 1989.  <p>   MATH2171, Analyse num&#233;rique 1a : depuis 1995 ( 50% en 1994)  <p>   MATH2180, Analyse num&#233;rique 2: depuis 1995, (50% en 1995-1998)  <p>   MATH2830, S&#233;minaire d'analyse num&#233;rique : depuis 1995, 33%  <p>   MATH2900, S&#233;minaire de math&#233;matique: 25% depuis 1998.  <p> 3<sup>&#232;<span class="roman">me</span></sup> cycle:  <p>   ``M&#233;thodes CF en approximation et th&#233;orie des syst&#232;mes'',    partie du cours de formation interdisciplinaire pour doctorands,                       Facult&#233;s N.D. de la Paix, Namur, janvier 1990.  <p>   Analyse num&#233;rique, special topics in approximation theory:  <ul> <li>  MAPA 3011A 1996-1997: CF methods in complex approximation<p></li>  <li>  MAPA 3072A 1997-1998: New difference operators &amp; related        orthogonal polynomials<p></li>  <li>  MAPA 3xxxA 1998-1999: Pad&#233; approximation to functions      with branch points.<p></li>  <li> MAPA3072A 1999-2000:   Semi-classical orthogonal polynomials on the unit circle.<p></li>  <li> MAPA 3071 2000-2001:   Asymptotic estimates in complex rational approximation.<p></li>  <li> MAPA 3xxx 2001-2002:   Asymptotic estimates in complex rational approximation.<p></li> </ul>  <p>   MAPA3116, S&#233;minaire d'analyse num&#233;rique : depuis 1995, 20%  <p>  Encadrement de th&#232;se (G.&nbsp;Bangerezako, dep<sup><span class="roman">t</span></sup> MATH):   50% 1997-1999.  <p>  <p> <br /><br /> <b>Co-&#233;dition de livre:</b>  <p> C.BREZINSKI , A.RONVEAUX , A.DRAUX , A.P.MAGNUS , P.MARONI, editors: <i>Polyn&#244;mes orthogonaux et applications Bar-le-Duc 1984 </i>, Lecture Notes in Mathematics <b>1171 </b>, Springer, Berlin 1985.  <p> <br /><br /> <table align="center" border="0"><tr><td> <font size="+2">Conf&#233;rences publi&#233;es et articles de p&#233;riodiques:</font></td></tr></table><!--hboxt-->  <p> <br /> <ul> <li>40    R.&nbsp;BOUCEKKINE, M.&nbsp;GERMAIN, O.&nbsp;LICANDRO, A.&nbsp;MAGNUS: Numerical solution by iterative methods of a class of vintage capital  models, <i>Journal of Economic Dynamics and Control</i>,    <b>25</b> nr 5 (Mai 2001) 655-669.<p></li>  <li>39   Lefevre  L, Dochain  D, de Azevedo  SF, Magnus  A:   Optimal selection of orthogonal polynomials applied to the    integration of chemical reactor equations by collocation methods   <i>COMPUTERS &amp; CHEMICAL ENGINEERING</i>    <b>24</b>: (12) 2571-2588 DEC 1 2000<p></li>  <li> Lef&#232;vre L., D. Dochain, A. Magnus (2000). "Numerical simulation of tubular reactors : some properties of the orthogonal collocation".  Proc. 3rd MATHMOD, I. Troch and F. Breitenecker (Eds.), pp.347-351.<p></li>  <li>38  A.P. Magnus, J. Meinguet, The elliptic functions and integrals of the `1/9' problem,  <i>Numerical Algorithms</i>  <b>24</b>: (1-2) (2000) 117-139.<p></li>  <li>37 G.&nbsp;BANGEREZAKO, A.P.&nbsp;MAGNUS, The factorization method for   the semi-classical polynomials, pp.&nbsp;295-300 in    <i>Self-Similar Systems</i>, edited by V.B.&nbsp;Priezzhev and    V.P.&nbsp;Spiridonov, Joint Institute for Nuclear Research, Dubna, 1999.<p></li>  <li> M.&nbsp;GERMAIN, A.P.&nbsp;MAGNUS, Progr&#232;s technique et dur&#233;e de    vie du capital,        Pr&#233;publication UCL: IRES Discussion paper 9911, 1999.   <a href="http://www.ires.ucl.ac.be/DP/Biblio_html/9911.html">http://www.ires.ucl.ac.be/DP/Biblio_html/9911.html</a><p></li>  <li>36 A.P.&nbsp;MAGNUS, Freud's equations for orthogonal polynomials as discrete Painlev&#233;     equations, pp. 228-243 in   <i>Symmetries and Integrability of Difference Equations</i>,         Edited by Peter A. Clarkson &amp; Frank W. Nijhoff,    Cambridge U.P., Lond. Math. Soc. Lect. Note Ser. 255, 1999. &lt;br&#62; See here the postscript preprint (complete, with the Chaucer lines!) <a href="freudpai.ps">freudpai.ps</a>  (162K)<p></li>  <li>35    R.&nbsp;BOUCEKKINE, M.&nbsp;GERMAIN, O.&nbsp;LICANDRO, A.&nbsp;MAGNUS:    Creative destruction, investment volatility, and the average age    of capital,    <i>  J. of Economic Growth</i> <b>3</b> (1998) 361-384.<p></li>  <li>   A.&nbsp;Magnus, L'&#233;ducation math&#233;matique du jeune Stendhal,    <i>Disquisitiones Mathematic&#230;</i> <b>1</b> (1998) n<sup><font face="symbol"></font ></sup>&nbsp;2,    2-6.  <a href="Stendhalmath.htm">Voir ici le fichier Stendhalmath.htm</a><p></li>  <li>34 Waldraff  W, Dochain  D, Bourrel  S, Magnus  A:   On the use of observability measures for sensor location in tubular reactor   <i>JOURNAL OF PROCESS CONTROL</i>   <b>8</b>: (5-6) 497-505 OCT-DEC 1998<p></li>  <li>33 M.GERMAIN, A.P.MAGNUS: L'impact du progr&#232;s technique     sur la croissance dans  un mod&#232;le     &#224; g&#233;n&#233;rations de capital,  <i>Cahiers d'&#233;conomie politique. Histoire de    la pens&#233;e et Th&#233;ories.</i> n<sup><font face="symbol"></font ></sup>&nbsp;32 (1998) 99-116.<p></li>  <li>32 A.P.&nbsp;MAGNUS, On optimal Pad&#233;-type cuts,  <i>Annals of Numerical Math.</i> <b>4</b> (1997) 435-450.<p></li>  <li>&nbsp;&nbsp; M.GERMAIN, A.P.MAGNUS: Consensus versus inflation dans le         cadre d'un mod&#232;le de Samuelson &#233;largi: &#233;quilibres,         cycles et chaos. <i>IRES Discussion Paper 9726, D&#233;pt.      sci. &#233;con. Univ. catholique Louvain</i>, Oct.&nbsp;1997.<p></li>  <li>31 A.P.&nbsp;MAGNUS, Painlev&#233; equations for semi-classical   recurrence coefficients: research problem 96-2, <i>Constructive    Approx.</i> <b>12</b> (1996) 303-306.  <a href="http://www.cecm.sfu.ca/personal/pborwein/CA_MOSAIC/PROBLEMS/P96-2.ps">postscript file</a><p></li>  <li>30 A.P.&nbsp;MAGNUS, Special non uniform lattice (<i>snul</i>)        orthogonal polynomials on discrete dense sets of points,        <i>J.&nbsp;Comp.&nbsp;Appl.&nbsp;Math. </i> <b>65 </b> (1995)        253-265.      <p>         Pr&#233;publication UCL: S&#233;minaire Math&#233;matique (nouvelle s&#233;rie)         n<sup><font face="symbol"></font ></sup>&nbsp;246,         Institut de Math&#233;matique Pure et Appliqu&#233;e UCL,         F&#233;vrier 1995.<p></li>  <li>29 A.P. MAGNUS Asymptotics for the simplest generalized     Jacobi polynomials recurrence coefficients from Freud's     equations: numerical explorations,     <i>Annals of Numerical Mathematics </i>     <b>2</b> (1995) 311-325.      <p>         Pr&#233;publication UCL: Recherches de math&#233;matique n<sup><font face="symbol"></font ></sup>&nbsp;&nbsp;40         Institut de Math&#233;matique Pure et Appliqu&#233;e UCL,         Juillet 1994.<p></li>  <li>28 A.P.MAGNUS:            Painlev&#233;-type differential equations for the recurrence            coefficients of semi-classical orthogonal polynomials.            <i>J.&nbsp;Comp.&nbsp;Appl.&nbsp;Math.</i> <b>57 </b> (1995) 215-237.<p></li>  <li>   M.GERMAIN, A.P.MAGNUS: De l'impact du progr&#232;s technique     sur la croissance dans      un mod&#232;le     &#224; g&#233;n&#233;rations de capital.        Pr&#233;publication UCL: IRES Discussion paper 9502, 1995.<p></li>  <li>27 M.&nbsp;GERMAIN, A.P.&nbsp;MAGNUS: Anticipations rationnelles et    apprentissage. De la pertinence des mod&#232;les coh&#233;rents,    <i>Recherches Economiques de Louvain  </i> <b>60 </b> (1994)    481-486.<p></li>  <li>26 A.P.MAGNUS: Asymptotics  and super asymptotics of      best rational approximation error norms      for the exponential function (the '1/9' problem)      by the Carath&#233;odory-Fej&#233;r's method,      pp.&nbsp;173-185 <i>in </i>      A.&nbsp;Cuyt, editor:      <i>Nonlinear Methods and Rational Approximation&nbsp;II </i>,      Kluwer, Dordrecht, 1994.<br />      <a href="antw93.ps">ps file 420K  </a>      <a href="antw93.pdf">pdf file 309K</a><p></li>  <li>25 A.P. MAGNUS Refined asymptotics for Freud's recurrence       coefficients, pp. 196-200 <i>in </i>       V.P.&nbsp;Havin &amp; N.K.&nbsp;Nikolski, editors:       <i>Linear and Complex Analysis Problem Book&nbsp;3, Part&nbsp;II </i>,        <i>Lecture Notes in Mathematics </i> <b>1574 </b>, Springer-Verlag,       Berlin, 1994.<p></li>  <li>24 T.ERDELYI, A.P.MAGNUS, P.NEVAI, ``Generalized Jacobi          weights, Christoffel functions and Jacobi polynomials.''          <i>SIAM J.&nbsp;Math.&nbsp;An.</i>  <b>25 </b> (1994)            602-614, 1461.<p></li>  <li>23 J.GILEWICZ, A.P.MAGNUS: ``Inverse Stieltjes iterates and          errors of Pad&#233; approximants in the whole complex plane.''          <i>J.&nbsp;Comp.&nbsp;Appl.&nbsp;Math.</i> <b>49 </b> (1993) 79-84.<p></li>  <li>22 J.GILEWICZ, A.P.MAGNUS: ``Optimal inequalities of Pad&#233;          approximant errors in the Stieltjes case: closing results.''          <i>Intern.&nbsp;Journal of Integral Transforms and Special Functions</i>         <b>1 </b> (1993) 9-18.  <p> <p></li>  <li>   A.P.MAGNUS: On Laplace transform and Darboux's method in the        investigation of Taylor series coefficients.        Pr&#233;publication UCL: S&#233;minaires Institut Math&#233;matique,        Rapport 198, octobre 1991.<p></li>  <li>21 A.P.MAGNUS: Towards quantitative results on Fibonacci        chain orthogonal polynomials, <i>The IMACS Annals on Computing        and Applied Mathematics </i> <b>9 </b> (1991)        (C.BREZINSKI, L.GORI &amp; A.RONVEAUX,        editors), Baltzer, Basel, 87-94.<p></li>  <li>20 J.GILEWICZ, A.P.MAGNUS: Sharp inequalities for the Pad&#233; approximant errors in the Stieltjes case, <i>Rocky Mount. J. Math. </i> <b>21 </b> (1991) 227-233.<p></li>  <li>19 W.VAN&nbsp;ASSCHE, A.P.MAGNUS: Sieved orthogonal polynomials and discrete measures with jumps dense in an interval, <i>Proc. AMS </i> <b>106 </b> (1989) 163-173.<p></li>  <li>18  A.P.MAGNUS: Associated Askey-Wilson polynomials as Laguerre-Hahn orthogonal polynomials, pp.261-278 <i>in </i> <i>Orthogonal Polynomials and their Applications, Proceedings, Segovia 1986 </i>, (Editors: M.Alfaro <i>et al. </i>), Lecture Notes in Mathematics <b>1329 </b>, Springer-Verlag  Berlin Heidelberg, 1988.<p></li>  <li>17  A.P.MAGNUS: On the use of Carath&#233;odory-Fej&#233;r method for investigating '1/9' and similar constants, pp. 105-132 <i>in </i> A.CUYT, Editor: <i>Nonlinear Numerical Methods and Rational Approximation </i>, D.Reidel, Dordrecht 1988.<p></li>  <li>16  A.P.MAGNUS: Toeplitz matrix techniques and convergence of complex weight Pad&#233; approximants, <i>J.Comp. Appl. Math. </i> <b>19 </b> (1987) 23-38.<p></li>  <li>15  A.P.MAGNUS, R.CHAUVAUX: Continuous spectrum in optical absorption for granular thin films, <i>Thin Solid Films </i> <b>142 </b> (1986) 279-308.<p></li>  <li>14  A.P.MAGNUS: On Freud's equations for exponential weights, <i>J. Approx. Th. </i> <b>46 </b> (1986) 65-99.<p></li>  <li>13  A.P.MAGNUS: A proof of Freud's conjecture about the orthogonal polynomials related to <font face="symbol">|</font ><i>x</i><font face="symbol">|</font ><sup><font face="symbol">a</font ></sup>exp(<font face="symbol">-</font ><i>x</i><sup>2<i>m</i></sup>) for integer <i>m</i>, pp. 362-372 <i>in </i> C.BREZINSKI , A.RONVEAUX , A.DRAUX , A.P.MAGNUS , P.MARONI editors: <i>Polyn&#244;mes orthogonaux et applications Bar-le-Duc 1984 </i>, Lecture Notes in Mathematics <b>1171 </b>, Springer, Berlin 1985.<p></li>  <li>12  A.P.MAGNUS, R.CHAUVAUX: On the discretisation of the double layer integral operator for surfaces of revolution, <i>J. Comp. Appl. Math. </i> <b>12&amp;13 </b> (1985) 467-473.<p></li>  <li>11  A.P.MAGNUS: Asymptotic behaviour of continued fraction coefficients related to singularities of the weight function, pp.22-45 <i>in </i> D.G.PETTIFOR, D.L.WEAIRE, editors: <i>The Recursion Method and its Applications </i>, Springer Series in Solid-State Sciences <b>58 </b>, Springer, Berlin 1985.<p></li>  <li>10  A.P.MAGNUS: Riccati acceleration of Jacobi continued fractions and Laguerre-Hahn orthogonal polynomials, pp 213-230 <i>in </i> H. WERNER , H.J. B&#220;NGER, editors: <i>Pad&#233; Approximation and its Applications, Bad Honnef 1983 </i>, Lecture Notes in Mathematics <b>1071 </b>, Springer Verlag , Berlin , 1984.<p></li>  <li>9  A.RONVEAUX, A.P.MAGNUS: Surface modes of a spherical void at the proximity of a surface, pp.123-128 <i>in </i> R.CAUDANO, J.M.GILLES and A.A.LUCAS, editors: <i>Vibrations at Surfaces </i>. Plenum Pub. Co., N.Y. 1982.<p></li>  <li>8  A.P.MAGNUS, Recurrence coefficients in case of Anderson localisation, pp.309-313 <i>in </i> H. van ROSSUM, M.G. de&nbsp;BRUIN,editors: <i>Pad&#233; Approximation and its Applications, Amsterdam 1980 </i>, Lecture Notes in Mathematics <b>888 </b>, Springer Verlag, Berlin, 1981.<p></li>  <li>7  A.P.MAGNUS: Rate of convergence of sequences of Pad&#233;-type approximants and pole detection in the complex plane, pp.300-308 <i>in </i> H. van ROSSUM,M.G.de BRUIN,editors: <i>Pad&#233; Approximation and its Applications, Amsterdam 1980 </i>, Lecture Notes in Mathematics <b>888 </b>, Springer Verlag, Berlin, 1981.<p></li>  <li>6  A.RONVEAUX, A.P.MAGNUS: ``Van der Waals energy between voids and particles. From asymptotic to close contact.'' <i>Solid State Comm. </i> <b>34 </b> (1980) 695-698.<p></li>  <li>5  A.P.MAGNUS: Recurrence coefficients for orthogonal polynomials on connected and non connected sets, pp. 150-171 <i>in </i> L.WUYTACK,editor: <i>Pad&#233; Approximation and its Applications </i>, Lecture Notes in Mathematics <b>765 </b>, Springer Verlag Berlin 1979.<p></li>  <li>4  J.GILEWICZ, A.P.MAGNUS: Valleys in c-table. pp.135-149 <i>in </i> L.WUYTACK,editor: <i>Pad&#233; Approximation and its Applications </i>, Lecture Notes in Mathematics <b>765 </b>, Springer Verlag Berlin 1979.<p></li>  <li>3  J.LEFEVRE, A.P.MAGNUS: Accuracy of simple difference-differential equations for blood flow in arteries. <i>Math. and Comp. in Simulation </i> <b>21 </b> (1979) 340-351.<p></li>  <li>2  Andr&#233; HAUTOT, A.P.MAGNUS: Calculation of the eigenvalues of Schr&#246;dinger equations by an extension of Hill's method,  <i>J. Comp. Appl. Math. </i> <b>5 </b> (1979) 3-15.<p></li>  <li>1  A.P.MAGNUS Fractions continues g&#233;n&#233;ralis&#233;es et matrices  infinies, <i>Bull. Soc. Math. Belg. </i> <b>29 </b> (ser.B) (1977) 145-159.<p></li> </ul>  <p>  <table align="center" border="0"><tr><td>  A para&#238;tre:</td></tr></table><!--hboxt-->  <p> <br /> <table align="center" border="0"><tr><td>  Soumis:</td></tr></table><!--hboxt-->  <p> <br />V. Pierrard, and A. Magnus, Lorentzian orthogonal polynomials  <p>  <p>  <p>  <p>  <table align="center" border="0"><tr><td>  En pr&#233;paration:</td></tr></table><!--hboxt-->  <p> <br /> <p>  <p>  <p>  <p>  <p> <br /><br /><b>S&#233;lection d'abstracts.</b>  <p> <br />   87h:42039 (18:11) 42C05      <p>    Magnus, Alphonse P. (B-UCL)     <p>    On Freud's equations for exponential weights. (English)  <p>    Papers dedicated to the memory of G&#233;za Freud.  <p>    Journal of Approximation Theory 46 (1986), no. 1, 65-99.  <p>    Geza Freud raised two conjectures that have generated considerable    interest in the last few years. Both of them are connected with the    recurrence coefficients for the orthonormal polynomials with respect    to the weights <span class="roman">exp</span>(<font face="symbol">-</font ><font face="symbol">|</font ><i>x</i><font face="symbol">|</font ><sup><font face="symbol">a</font ></sup>), <font face="symbol">a</font > &gt; 0. The    first conjecture was settled by A. A. Rachmanov while the second one    has resisted every attempt at solution for several years. This    conjecture claims that if the recurrence relation is <i>xp</i><sub><i>n</i></sub>(<i>x</i>)=<i>a</i><sub><i>n</i>+1</sub><i>p</i><sub><i>n</i>+1</sub>(<i>x</i>)+<i>b</i><sub><i>n</i></sub><i>p</i><sub><i>n</i></sub>(<i>x</i>)+<i>a</i> <sub><i>n</i></sub><i>p</i><sub><i>n</i><font face="symbol">-</font >1</sub>(<i>x</i>), then (*)    <br clear="all" /><table border="0" align="left"><tr><td nowrap="nowrap"></td><td nowrap="nowrap" align="center"> <font size="-1"></font><!--sup --><br />lim<br /> <font size="-1"><i>n</i><font face="symbol"></font ><font face="symbol"></font ></font>&nbsp;<br /></td><td nowrap="nowrap" align="center"> <i>a</i><sub><i>n</i></sub>/<i>n</i><sup>1/<font face="symbol">a</font ></sup> </td></tr></table><br /> exists (if it    exists then the limit is easy to compute). The paper under review    solves Freud's conjecture in the important special case when <font face="symbol">a</font >    is an even integer; more precisely, the existence of (*) is proved    for weights of the form exp(<font face="symbol">-</font ><i>P</i>(<i>x</i>)) where <i>P</i> is a polynomial of    degree 2<i>m</i>, <i>m</i>=1,2,<font face="symbol"></font >, with positive leading coefficient. The    technique of the proof is very fine; it starts from an identity of    Freud for the <i>a</i><sub><i>n</i></sub>'s.  <p>    It should be mentioned that by a different    potential-approximation-theoretic approach Freud's conjecture has    recently been fully settled by D. S. Lubinsky, H. N. Mhaskar and E. B.    Saff.  <p>    See also the following review.  <p>    {For the entire collection see MR 87a:41003.}  <p>              Reviewed by Totik, V. (Szeged)  <p>    Cited in: 88d:42039  <p>     Copyright American Mathematical Society 1987, 1995  <p>  <table align="center" border="0"><tr><td> **************************************************</td></tr></table><!--hboxt-->  <p> <br />   88i:65022 (19:16) 65D15   <p>    Magnus, Alphonse P. (B-UCL)      <p>    Toeplitz matrix techniques and convergence of complex weight Pade    approximants. (English)       <p>    Journal of Computational and Applied Mathematics 19 (1987), no. 1,    23-38.  <p>    Starting from the Stieltjes (or Markov) class of functions, the author    establishes convergence for diagonal Pade approximants given a choice    of comparison weight functions. This class of functions may be    generated from spectral investigations of selfadjoint operators. The    paper links the convergence established above with that of projection    methods of operators and thereafter utilizes well-established results    in Toeplitz operator theory, thus providing interesting insight into    the linkages between Pade approximants and special operators.  <p>              Reviewed by Rodrigues, A. J. (Nairobi)  <p>     Copyright American Mathematical Society 1988, 1995  <p>  <table align="center" border="0"><tr><td> **************************************************</td></tr></table><!--hboxt-->  <p> <br /> A.P.MAGNUS: On the use of Carath&#233;odory-Fej&#233;r method for investigating '1/9' and similar constants, pp. 105-132 <i>in </i> A.CUYT, Editor: <i>Nonlinear Numerical Methods and Rational Approximation </i>, D.Reidel, Dordrecht 1988.  <p>   Let <i>E</i><sub><i>n</i></sub> be the error norm of the best <i>L</i><sub><font face="symbol"></font ></sub> rational approximation   of degree <i>n</i> to the exponential function exp(<font face="symbol">-</font ><i>t</i>) on [0,<font face="symbol"></font >).   Grounds are given for setting the conjectured       limit <i>E</i><sub><i>n</i></sub>/<i>q</i><sup><i>n</i></sup><font face="symbol"></font > 2<i>q</i><sup>1/2</sup> when <i>n</i><font face="symbol"></font ><font face="symbol"></font >, where <i>q</i> is    the known constant `1/9'=    1/9.2890254919208189187554494359517450610316948677<font face="symbol"></font >,    based on the singular values and functions of the relevant    Hankel operator (Carath&#233;odory-Fej&#233;r's method).    Moreover, hints are given according to which a valuable asymptotic   expansion of <i>E</i><sub><i>n</i></sub> should also contain <i>n</i><sup><span class="roman">th</span></sup> powers of new   constants <i>q</i><sub>1</sub>=`1/56<font face="symbol"></font >, <i>q</i><sub>2</sub>=`1/240<font face="symbol"></font >, etc.  <p>  A vrai dire, cet article-partie d'actes est assez peu cit&#233;, on cite plut&#244;t  <p>    (cf.    A.A.&nbsp;GONCHAR, E.A.&nbsp;RAKHMANOV, Equilibrium distribution and      the degree of rational approximation of analytic functions,     <i>Mat.&nbsp;Sb.&nbsp;</i> <b>134 </b> (176) (1987) 306-352 =     <i>Math.&nbsp;USSR Sbornik </i> <b>62 </b> (1989) 305-348,  <p>        R.S.&nbsp;VARGA, <i>Scientific Computation on Mathematical         Problems and Conjectures, </i> CBMS-NSF Reg.&nbsp;Conf.&nbsp;Series in         Appl.&nbsp;Math.&nbsp;<b>60 </b>, SIAM, Philadelphia, 1990)  <p> une lettre que j'ai envoy&#233;e &#224; une dizaine de personnes... Les russes, prudents avec les questions ethniques, ont jug&#233; bon de la r&#233;f&#233;rencer comme ``Preprint B-1348, Inst. Math., Katholieke Univ. Leuven, Louvain, 1986''.  <p>  <table align="center" border="0"><tr><td> **************************************************</td></tr></table><!--hboxt-->  <p> <br />   95f:33011b (94:16) 33C45 26C05 42C05   <p>    Erd&#233;lyi, Tamas (3-SFR) ; Magnus, Alphonse P. (B-UCL-PA) ; Nevai, Paul    (1-OHS)   <p>     ``Generalized Jacobi weights, Christoffel functions, and    Jacobi polynomials''. (English)          <i>SIAM J.&nbsp;Math.&nbsp;An.</i>  <b>25 </b> (1994)            602-614.   <p>    Erratum: SIAM Journal on Mathematical Analysis 25 (1994), no. 5, 1461.  <p>    It is well known that the Legendre polynomial <i>P</i><sub><i>n</i></sub>(<i>x</i>) satisfies    the Bernstein inequality  <br clear="all" /><table border="0" width="100%"><tr><td> <table align="center"><tr><td nowrap="nowrap" align="center"> (sin<font face="symbol">q</font >)<sup>1/2</sup> <font face="symbol">|</font ><i>P</i><sub><i>n</i></sub>(cos<font face="symbol">q</font >)<font face="symbol">|</font > &lt;  (2/<font face="symbol">p</font >)<sup>1/2</sup> <i>n</i><sup><font face="symbol">-</font >1/2</sup>,&nbsp;&nbsp;&nbsp; 0  <font face="symbol"></font > <font face="symbol">q</font >  <font face="symbol"></font > <font face="symbol">p</font >.</td></tr></table> </td></tr></table>   V. A. Antonov and K. V. Kholshevnikov [Vestnik Leningrad.    Univ. Mat. Mekh. Astronom. 1980, no. 3, 5-7, 128; MR 82b:33012]    obtained a sharper result by showing that the factor <i>n</i><sup><font face="symbol">-</font >1/2</sup> can    be replaced by (<i>n</i>+1/2)<sup><font face="symbol">-</font >1/2</sup>. For ultraspherical polynomials    <i>P</i><sub><i>n</i></sub><sup>(<font face="symbol">l</font >)</sup>(<i>x</i>), L. Lorch [Appl. Anal. 14 (1982/83), no. 3,    237-240; MR 84k:26017] proved that  <br clear="all" /><table border="0" width="100%"><tr><td> <table align="center"><tr><td nowrap="nowrap" align="center"> (sin<font face="symbol">q</font >)<sup><font face="symbol">l</font ></sup> <font face="symbol">|</font > <i>P</i><sub><i>n</i></sub><sup>(<font face="symbol">l</font >)</sup>(cos<font face="symbol">q</font >)<font face="symbol">|</font > &lt;  2<sup>1<font face="symbol">-</font ><font face="symbol">l</font ></sup> <font face="symbol">G</font >(<font face="symbol">l</font >)<sup><font face="symbol">-</font >1</sup> (<i>n</i>+<font face="symbol">l</font >)<sup><font face="symbol">l</font ><font face="symbol">-</font >1</sup></td></tr></table> </td></tr></table>   for 0 &lt; <font face="symbol">l</font > &lt; 1    and 0 <font face="symbol"></font > <font face="symbol">q</font > <font face="symbol"></font > <font face="symbol">p</font >.  <p>    Recently, Y. H. Chou, L. Gatteschi and R. Wong [Proc. Amer. Math. Soc.    121 (1994), no. 3, 703-709; MR 94i:33008] have extended this result    to nonsymmetric Jacobi polynomials <i>P</i><sub><i>n</i></sub><sup>(<font face="symbol">a</font >,<font face="symbol">b</font >)</sup>(<i>x</i>)    with <font face="symbol">-</font >1/2  &lt;  <font face="symbol">a</font >, <font face="symbol">b</font > &lt;  1/2, and <font face="symbol">a</font >+<font face="symbol">b</font > &gt; 0.  <p>    Here, in this interesting paper, a more general result is obtained.    For orthonormal Jacobi polynomials <i>p</i><sub><i>n</i></sub>(<i>w</i>,<i>x</i>) with respect to the    weight <i>w</i>(<i>x</i>)=(1<font face="symbol">-</font ><i>x</i>)<sup><font face="symbol">a</font ></sup>(1+<i>x</i>)<sup><font face="symbol">b</font ></sup> with <font face="symbol">a</font > <font face="symbol"></font > <font face="symbol">-</font >1/2 and    <font face="symbol">b</font > <font face="symbol"></font > <font face="symbol">-</font >1/2, the inequality  <br clear="all" /><table border="0" width="100%"><tr><td> <table align="center"><tr><td nowrap="nowrap" align="center">  </td><td nowrap="nowrap" align="center"> <font size="-1"></font><!--sup --><br />max<br /> <font size="-1"><i>x</i>  <font face="symbol"></font > [<font face="symbol">-</font >1,1]</font>&nbsp;<br /></td><td nowrap="nowrap" align="center"> </td><td align="left" class="cl"><br /><font face="symbol"><font size="+2"></font><br /></font> <div class="comb">&nbsp;</div> </td><td nowrap="nowrap" align="center">  <div class="hrcomp"><hr noshade="noshade" size="1"/></div> <div class="norm">1<font face="symbol">-</font ><i>x</i><sup>2</sup><br /></div> <div class="comb">&nbsp;</div> </td><td nowrap="nowrap" align="center"> <i>w</i>(<i>x</i>) <i>p</i><sub><i>n</i></sub>(<i>w</i>,<i>x</i>)<sup>2</sup>  <font face="symbol"></font > </td><td nowrap="nowrap" align="center"> <table border="0"><tr><td nowrap="nowrap" align="center"> 2<i>e</i>(2+</td><td align="left" class="cl"><br /><font face="symbol"><font size="+2"></font><br /></font> <div class="comb">&nbsp;</div> </td><td nowrap="nowrap" align="center">  <div class="hrcomp"><hr noshade="noshade" size="1"/></div> <div class="norm"><font face="symbol">a</font ><sup>2</sup>+<font face="symbol">b</font ><sup>2</sup><br /></div> <div class="comb">&nbsp;</div> </td><td nowrap="nowrap" align="center"> )</td></tr></table> <div class="hrcomp"><hr noshade="noshade" size="1"/></div><font face="symbol">p</font ><br /></td><td nowrap="nowrap" align="center"> </td></tr></table> </td></tr></table>   holds for every <i>n</i> <font face="symbol"></font > 0. The authors,    although unable to obtain sharp constants, develop here techniques    which are themselves very important. They use generalized algebraic    polynomials to estimate the Christoffel functions <font face="symbol">l</font ><sub><i>n</i></sub>(<i>w</i>,<i>x</i>)    and obtain a Riccati equation which yields estimates for <i>p</i><sub><i>n</i></sub>(<i>w</i>,<i>x</i>)<sup>2</sup><font face="symbol">l</font ><sub><i>n</i></sub>(<i>w</i>,<i>x</i>).  <p>              Reviewed by Guadalupe Hern&#225;ndez, Jos&#233; Javier (Logrono)  <p>     Copyright American Mathematical Society 1995  <p>  <table align="center" border="0"><tr><td> **************************************************</td></tr></table><!--hboxt-->  <p> A.&nbsp;P.&nbsp;MAGNUS:            Freud's equations for orthogonal polynomials                  as discrete Painlev&#233; equations.  <p> We consider orthogonal polynomials <i>p</i><sub><i>n</i></sub> with respect to an exponential  weight function  <i>w</i>(<i>x</i>)=exp(<font face="symbol">-</font ><i>P</i>(<i>x</i>)). The related equations for the recurrence coefficients have been explored by many people, starting essentially with Laguerre, in order to study special continued fractions, recurrence relations, and various asymptotic expansions (G.&nbsp;Freud's contribution).  <p> Most striking example is <i>n</i>=2<i>tw</i><sub><i>n</i></sub>+<i>w</i><sub><i>n</i></sub>(<i>w</i><sub><i>n</i>+1</sub>+<i>w</i><sub><i>n</i></sub>+<i>w</i><sub><i>n</i><font face="symbol">-</font >1</sub>) for the recurrence coefficients <i>p</i><sub><i>n</i>+1</sub>=<i>xp</i><sub><i>n</i></sub><font face="symbol">-</font ><i>w</i><sub><i>n</i></sub> <i>p</i><sub><i>n</i><font face="symbol">-</font >1</sub> of the orthogonal polynomials related to the weight <i>w</i>(<i>x</i>)=exp(<font face="symbol">-</font >4(<i>tx</i><sup>2</sup>+<i>x</i><sup>4</sup>)).  This example appears in practically all the references. The connection with discrete Painlev&#233; equations is described here.  <p>  <table align="center" border="0"><tr><td> **************************************************</td></tr></table><!--hboxt-->  <p> <br /><br /><b>Activit&#233;s d'&#233;dition.</b>  <p> Je suis membre de l'editorial board du <i>Journal of  Approximation Theory</i>.  <p> <br /><br /><b>Collaboration &#224; l'organisation de congr&#232;s.</b>  <p> <i>Polyn&#244;mes orthogonaux et applications</i>,  Bar-le-Duc,  15-18 octobre 1984.  <p> <br /><br /><b>Invitations  &#224; des conf&#233;rences et congr&#232;s.</b>  <p> <i>Recursion Conference and Applications</i>, Imperial College, London, 13 &amp; 14 septembre 1984.  <p> <i>Third International Symposium on Orthogonal Polynomials and their Applications</i>, Erice (Trapani), 1-8 juin 1990.  <p> <i>Constructive Methods in Complex Analysis</i>, Oberwolfach, 26 mars-1 avril 1995.  <p> <i>Symmetries and Integrability of Difference Equations II</i>, Canterbury, 1-5 juillet  1996.  <p> <br /><br /><b>Pr&#233;sences  &#224; des conf&#233;rences et congr&#232;s.</b>  <p> <br /> <i>Pad&#233; Approximation and its Applications </i>, Antwerpen 1979.  <p> <i>Pad&#233; Approximation and its Applications </i>, Amsterdam 1980.  <p>  <i>Pad&#233; Approximation and its Applications </i>, Bad Honnef 1983.  <p>  <i>Pad&#233; Approximation and its Applications </i>,        Marseille-Luminy, 14-18 October 1985.  <p> <i>Nonlinear Numerical Methods and Rational Approximation </i>, Antwerpen,   1987.  <p> <i>2<sup><span class="roman">nd</span></sup> International Symposium on Orthogonal Polynomials  and their Applications </i>, Segovia 1988.  <p> <i>4<sup><span class="roman">th</span></sup> International Symposium on Orthogonal Polynomials  and their Applications </i>, Evian 1992.  <p> <i>Nonlinear Numerical Methods and Rational Approximation II </i>, Antwerpen,   1993.  <p> <i>Computation in Economics, Finance and Engineering: Economic  Systems</i>, Cambridge (GB), 1998.  <p> <i>International conference on rational approximation</i>, Anvers,  6-11 juin 1999.  <p>  <br /><br /><hr /><small>File translated from T<sub><font size="-1">E</font></sub>X by <a href="http://hutchinson.belmont.ma.us/tth/"> T<sub><font size="-1">T</font></sub>H</a>, version 3.11.<br />On 18 Jun 2002, 14:27.</small> </html> 
