Alphonse P. Magnus

Adresse

 bureau:

 Université Catholique de Louvain

 Institut Mathématique

 Chemin du Cyclotron,2

 1348 Louvain-la-Neuve.

 e-mail  alphonse.magnus@uclouvain.be

Titres

Ingénieur civil en mathématiques appliquées (grade scientifique), UCL, 1969.

Docteur en sciences appliquées (grade scientifique), UCL, 1976.

Nominations

Assistant, UCL, 1969,

Premier assistant, UCL, 1976,

Chef de travaux, UCL, 1984.

Chargé de cours à temps partiel, UCL, 1997.

Professeur à temps partiel, 2004.

Discipline

Mathématiques, Analyse numérique, Théorie de l'approximation.

Enseignement, encadrement:

Encadrement exercices analyse numérique en MATH21, MAP21, MAP22 et PHYS 21-22 (aussi INFO21 ou 22 1980-1990).

MATH2171, Analyse numérique 1a : depuis 1972

MATH2172, Analyse numérique 1b : 1972-1994

Encadrement cours et exercices analyse FSA11 et 12: quelques intérims,

Encadrement exercices programmation FSA11 et/ou MATH12 : période 1976-1980.

Directions régulières de mémoires MAP23 et MATH22.

Cours

2^ème cycle:

``Approximation complexe quantitative'', cours de questions spéciales en mathématique,

Facultés N.D. de la Paix, Namur, janvier-mai 1989.

MATH2171, Analyse numérique 1a : depuis 1995 ( 50% en 1994)

MATH2180, Analyse numérique 2: depuis 1995, (50% en 1995-1998)

MATH2830, Séminaire d'analyse numérique : depuis 1995, 33% (50% depuis 2004).

MATH2900, Séminaire de mathématique: 25% depuis 1998.

INMA 2375, Projet intégré en ingénierie mathématique: 14% 2002-2004.

3^ème cycle:

``Méthodes CF en approximation et théorie des systèmes'', partie du cours de formation interdisciplinaire pour doctorands, Facultés N.D. de la Paix, Namur, janvier 1990.

Analyse numérique, special topics in approximation theory:

• MAPA 3011A 1996-1997: CF methods in complex approximation

• MAPA 3072A 1997-1998: New difference operators & related orthogonal polynomials

• MAPA 3xxxA 1998-1999: Padé approximation to functions with branch points.

• MAPA3072A 1999-2000: Semi-classical orthogonal polynomials on the unit circle.

• MAPA 3071 2000-2001: Asymptotic estimates in complex rational approximation.

• MAPA 3xxx 2001-2002: Asymptotic estimates in complex rational approximation.

• MAPA3071 2002-2003

• etc.

MAPA3116, Séminaire d'analyse numérique : 1995-2001, 20%

MAPA3079 Séminaire d'analyse numérique (DEA Math.), depuis 2002, 11%.

Encadrement de thèse (G. Bangerezako, dep^t MATH): 50% 1997-1999.

Co-édition de livre:

C.BREZINSKI , A.RONVEAUX , A.DRAUX , A.P.MAGNUS , P.MARONI, editors: Polynômes orthogonaux et applications Bar-le-Duc 1984 , Lecture Notes in Mathematics 1171 , Springer, Berlin 1985.

Conférences publiées et articles de périodiques:

---

• 52 Alphonse P. Magnus, François Ndayiragije, André Ronveaux, About families of orthogonal polynomials satisfying Heun's differential equation, Journal of Approximation Theory, 263 (2021) 105522,
  https://doi.org/10.1016/j.jat.2020.105522
  The last preprint.

 

• Why did we live almost 200 years with the wrong face of Legendre? BMS-NCM Newsletter #121, January 15, 2019, 11-12. http://bms.ulb.ac.be/BMSNCM/BMSNE121.pdf

 

• 51 A.P.Magnus, Gaussian integration formulas for logarithmic weights and application to 2-dimensional solid-state lattices
  Journal of Approximation Theory, 228 (2018) 21-57
  https://doi.org/10.1016/j.jat.2018.02.001,    The last preprint.
  

• 50 J.F. Fagnart, M. Germain, A. Magnus, Strong sustainability, rent and value-added sharing, Annals of Economics and Statistics, nr 121/122, June 2016, pp. 309-358.

• J.F. Fagnart, M. Germain, A. Magnus, Soutenabilité forte, rente et partage de la valeur ajoutée, No 2013021, Discussion Papers (IRES - Institut de Recherches Economiques et Sociales) from Université catholique de Louvain, Institut de Recherches Economiques et Sociales (IRES)

• 49 M.Germain, H.Tulkens, A.Magnus, Dynamic core-theoretic cooperation in a two-dimensional international environmental model, Mathematical Social Sciences 59 (2010) 208-226 = CORE discussion paper 2009/21, http://www.uclouvain.be/cps/ucl/doc/core/documents/coredp2009_21.pdf

• 48 Stanislaw Tokarzewski, Alphonse Ph. Magnus, Jacek Gilewicz, Estimation of a Stieltjes function expanded to Taylor series at complex conjugate points, Journal of Computational and Applied Mathematics Volume 233, Issue 3, (1 December 2009) Pages 835-841 http://dx.doi.org/10.1016/j.cam.2009.02.085

• 47 Alphonse P. Magnus, Rational interpolation to solutions of Riccati difference equations on elliptic lattices, Journal of Computational and Applied Mathematics, 233 (Issue 3, 1 Dec. 2009, 9th OPSFA Conference ) 793-801. Available online: 26-FEB-2009, http://dx.doi.org/10.1016/j.cam.2009.02.047

• 46 Alphonse P. Magnus, Elliptic Hypergeometric Solutions to Elliptic Difference Equations, SIGMA 5 (March 2009), 038, 12 pages , http://www.emis.de/journals/SIGMA/2009/038/sigma09-038.pdf

• 45 Alphonse P. Magnus, Viviane Pierrard, Formulas for recurrence coefficients of orthogonal polynomials related to Lorentzian-like weights, Journal of Computational and Applied Mathematics, Volume 219, Issue 2, Pages 431-440 (1 October 2008) , Special Issue dedicated to Prof. Claude Brezinski, on the occasion of his retirement from the University of Sciences and Technologies of Lille in June 2006, Edited by M. Prévost, M. Redivo-Zaglia and F. Wielonsky .
  The pdf preprint, with the % in the acknowledgements!

• 44 Germain, M.; Magnus, A.; Steenberghe, V., How to design and use the clean development mechanism under the Kyoto Protocol? A developing country perspective, Environmental and Resource Economics , Volume 38, Number 1, September 2007, pp. 13-30.
  The mathematical appendix (pdf 125K)

• 43 Marc Germain and Alphonse Magnus, Prices versus quantities: Stock pollution control with repeated choice of the instrument, Journal of Computational and Applied Mathematics Volume 197, Issue 2 , 15 December 2006, Pages 437-445

• 42 Alphonse P. Magnus, Freud equations for Legendre polynomials on a circular arc and solution of the Grünbaum-Delsarte-Janssen-Vries problem, Journal of Approximation Theory, Volume 139, Issues 1-2, This Volume is Dedicated to Barry Simon on the Occasion of his Sixtieth Birthday, March-April 2006, Pages 75-90.

• 41 Marc Germain; Vincent Van Steenberghe; Alphonse Magnus, Optimal Policy with Tradable and Bankable Pollution Permits: Taking the Market Microstructure into Account, Journal of Public Economic Theory 6 (2004) pp. 737-757.

• 40 R. BOUCEKKINE, M. GERMAIN, O. LICANDRO, A. MAGNUS: Numerical solution by iterative methods of a class of vintage capital models, Journal of Economic Dynamics and Control, 25 nr 5 (Mai 2001) 655-669.

• 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 COMPUTERS & CHEMICAL ENGINEERING 24: (12) 2571-2588 DEC 1 2000

• Lefèvre 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.

• 38 A.P. Magnus, J. Meinguet, The elliptic functions and integrals of the `1/9' problem, Numerical Algorithms 24: (1-2) (2000) 117-139.

• 37 G. BANGEREZAKO, A.P. MAGNUS, The factorization method for the semi-classical polynomials, pp. 295-300 in Self-Similar Systems, edited by V.B. Priezzhev and V.P. Spiridonov, Joint Institute for Nuclear Research, Dubna, 1999.

• M. GERMAIN, A.P. MAGNUS, Progrès technique et durée de vie du capital, Prépublication UCL: IRES Discussion paper 9911, 1999. http://www.ires.ucl.ac.be/DP/Biblio_html/9911.html

• 36 A.P. MAGNUS, Freud's equations for orthogonal polynomials as discrete Painlevé equations, pp. 228-243 in Symmetries and Integrability of Difference Equations, Edited by Peter A. Clarkson & Frank W. Nijhoff, Cambridge U.P., Lond. Math. Soc. Lect. Note Ser. 255, 1999. <br> See here the pdf preprint (complete, with the Chaucer lines!) freudpain.pdf (217K)

• 35 R. BOUCEKKINE, M. GERMAIN, O. LICANDRO, A. MAGNUS: Creative destruction, investment volatility, and the average age of capital, J. of Economic Growth 3 (1998) 361-384.

• A. Magnus, L'éducation mathématique du jeune Stendhal, Disquisitiones Mathematicæ 1 (1998) n^° 2, 2-6. Voir ici le fichier Stendhalmath.htm

• 34 Waldraff W, Dochain D, Bourrel S, Magnus A: On the use of observability measures for sensor location in tubular reactor JOURNAL OF PROCESS CONTROL 8: (5-6) 497-505 OCT-DEC 1998

• 33 M.GERMAIN, A.P.MAGNUS: L'impact du progrès technique sur la croissance dans un modèle à générations de capital, Cahiers d'économie politique. Histoire de la pensée et Théories. n^° 32 (1998) 99-116.

• 32 A.P. MAGNUS, On optimal Padé-type cuts, Annals of Numerical Math. 4 (1997) 435-450. pdf file 256K

•    M.GERMAIN, A.P.MAGNUS: Consensus versus inflation dans le cadre d'un modèle de Samuelson élargi: équilibres, cycles et chaos. IRES Discussion Paper 9726, Dépt. sci. écon. Univ. catholique Louvain, Oct. 1997.

• 31 A.P. MAGNUS, Painlevé equations for semi-classical recurrence coefficients: research problem 96-2, Constructive Approx. 12 (1996) 303-306. postscript file

• 30 A.P. MAGNUS, Special non uniform lattice (snul) orthogonal polynomials on discrete dense sets of points, J. Comp. Appl. Math. 65 (1995) 253-265.

  Prépublication UCL: Séminaire Mathématique (nouvelle série) n^° 246, Institut de Mathématique Pure et Appliquée UCL, Février 1995.

• 29 A.P. MAGNUS Asymptotics for the simplest generalized Jacobi polynomials recurrence coefficients from Freud's equations: numerical explorations, Annals of Numerical Mathematics 2 (1995) 311-325.
  

  Prépublication UCL: Recherches de mathématique n^°  40 Institut de Mathématique Pure et Appliquée UCL, Juillet 1994.
  gjacobimagnus.pdf The pdf preprint (128K)

• 28 A.P.MAGNUS: Painlevé-type differential equations for the recurrence coefficients of semi-classical orthogonal polynomials. J. Comp. Appl. Math. 57 (1995) 215-237. pdf file 300K

• M.GERMAIN, A.P.MAGNUS: De l'impact du progrès technique sur la croissance dans un modèle à générations de capital. Prépublication UCL: IRES Discussion paper 9502, 1995.

• 27 M. GERMAIN, A.P. MAGNUS: Anticipations rationnelles et apprentissage. De la pertinence des modèles cohérents, Recherches Economiques de Louvain 60 (1994) 481-486.

• 26 A.P.MAGNUS: Asymptotics and super asymptotics of best rational approximation error norms for the exponential function (the '1/9' problem) by the Carathéodory-Fejér's method, pp. 173-185 in A. Cuyt, editor: Nonlinear Methods and Rational Approximation II , Kluwer, Dordrecht, 1994.
  ps file 420K pdf file 309K

• 25 A.P. MAGNUS Refined asymptotics for Freud's recurrence coefficients, pp. 196-200 in V.P. Havin & N.K. Nikolski, editors: Linear and Complex Analysis Problem Book 3, Part II , Lecture Notes in Mathematics 1574 , Springer-Verlag, Berlin, 1994.
  pdf preprint 2.5Mb

• 24 T.ERDELYI, A.P.MAGNUS, P.NEVAI, ``Generalized Jacobi weights, Christoffel functions and Jacobi polynomials.'' SIAM J. Math. An. 25 (1994) 602-614, 1461.
  pdf preprint

• 23 J.GILEWICZ, A.P.MAGNUS: ``Inverse Stieltjes iterates and errors of Padé approximants in the whole complex plane.'' J. Comp. Appl. Math. 49 (1993) 79-84.

• 22 J.GILEWICZ, A.P.MAGNUS: ``Optimal inequalities of Padé approximant errors in the Stieltjes case: closing results.'' Intern. Journal of Integral Transforms and Special Functions 1 (1993) 9-18.

• A.P.MAGNUS: On Laplace transform and Darboux's method in the investigation of Taylor series coefficients. Prépublication UCL: Séminaires Institut Mathématique, Rapport 198, octobre 1991.

• 21 A.P.MAGNUS: Towards quantitative results on Fibonacci chain orthogonal polynomials, The IMACS Annals on Computing and Applied Mathematics 9 (1991) (C.BREZINSKI, L.GORI & A.RONVEAUX, editors), Baltzer, Basel, 87-94.

• 20 J.GILEWICZ, A.P.MAGNUS: Sharp inequalities for the Padé approximant errors in the Stieltjes case, Rocky Mount. J. Math. 21 (1991) 227-233.

• 19 W.VAN ASSCHE, A.P.MAGNUS: Sieved orthogonal polynomials and discrete measures with jumps dense in an interval, Proc. AMS 106 (1989) 163-173.

• 18 A.P.MAGNUS: Associated Askey-Wilson polynomials as Laguerre-Hahn orthogonal polynomials, pp.261-278 in Orthogonal Polynomials and their Applications, Proceedings, Segovia 1986 , (Editors: M.Alfaro et al. ), Lecture Notes in Mathematics 1329 , Springer-Verlag Berlin Heidelberg, 1988. pdf 2000K

• 17 A.P.MAGNUS: On the use of Carathéodory-Fejér method for investigating '1/9' and similar constants, pp. 105-132 in A.CUYT, Editor: Nonlinear Numerical Methods and Rational Approximation , D.Reidel, Dordrecht 1988. pdf file 385K

• 16 A.P.MAGNUS: Toeplitz matrix techniques and convergence of complex weight Padé approximants, J.Comp. Appl. Math. 19 (1987) 23-38.

• 15 A.P.MAGNUS, R.CHAUVAUX: Continuous spectrum in optical absorption for granular thin films, Thin Solid Films 142 (1986) 279-308.

• A.P. Magnus, CFGT determination of Varga's constant '1/9', sent to several people in October 1985, and been given to all the participants of the Padé approximation meeting organized by C. Brezinski at Marseille-Luminy, 14-18 Oct. 1985.

  September 2000 retyping, cf85.pdf the pdf file.

• 14 A.P.MAGNUS: On Freud's equations for exponential weights, J. Approx. Th. 46 (1986) 65-99.

• 13 A.P.MAGNUS: A proof of Freud's conjecture about the orthogonal polynomials related to |x|^aexp(-x^2m) for integer m, pp. 362-372 in C.BREZINSKI , A.RONVEAUX , A.DRAUX , A.P.MAGNUS , P.MARONI editors: Polynômes orthogonaux et applications Bar-le-Duc 1984 , Lecture Notes in Mathematics 1171 , Springer, Berlin 1985.

• 12 A.P.MAGNUS, R.CHAUVAUX: On the discretisation of the double layer integral operator for surfaces of revolution, J. Comp. Appl. Math. 12&13 (1985) 467-473.

• 11 A.P.MAGNUS: Asymptotic behaviour of continued fraction coefficients related to singularities of the weight function, pp.22-45 in D.G.PETTIFOR, D.L.WEAIRE, editors: The Recursion Method and its Applications , Springer Series in Solid-State Sciences 58 , Springer, Berlin 1985.

• 10 A.P.MAGNUS: Riccati acceleration of Jacobi continued fractions and Laguerre-Hahn orthogonal polynomials, pp 213-230 in H. WERNER , H.J. BÜNGER, editors: Padé Approximation and its Applications, Bad Honnef 1983 , Lecture Notes in Mathematics 1071 , Springer Verlag , Berlin , 1984.

• 9 A.RONVEAUX, A.P.MAGNUS: Surface modes of a spherical void at the proximity of a surface, pp.123-128 in R.CAUDANO, J.M.GILLES and A.A.LUCAS, editors: Vibrations at Surfaces . Plenum Pub. Co., N.Y. 1982.

• 8 A.P.MAGNUS, Recurrence coefficients in case of Anderson localisation, pp.309-313 in H. van ROSSUM, M.G. de BRUIN,editors: Padé Approximation and its Applications, Amsterdam 1980 , Lecture Notes in Mathematics 888 , Springer Verlag, Berlin, 1981.

• 7 A.P.MAGNUS: Rate of convergence of sequences of Padé-type approximants and pole detection in the complex plane, pp.300-308 in H. van ROSSUM,M.G.de BRUIN,editors: Padé Approximation and its Applications, Amsterdam 1980 , Lecture Notes in Mathematics 888 , Springer Verlag, Berlin, 1981.

• 6 A.RONVEAUX, A.P.MAGNUS: ``Van der Waals energy between voids and particles. From asymptotic to close contact.'' Solid State Comm. 34 (1980) 695-698.

• 5 A.P.MAGNUS: Recurrence coefficients for orthogonal polynomials on connected and non connected sets, pp. 150-171 in L.WUYTACK,editor: Padé Approximation and its Applications , Lecture Notes in Mathematics 765 , Springer Verlag Berlin 1979.

• 4 J.GILEWICZ, A.P.MAGNUS: Valleys in c-table. pp.135-149 in L.WUYTACK,editor: Padé Approximation and its Applications , Lecture Notes in Mathematics 765 , Springer Verlag Berlin 1979.

• 3 J.LEFEVRE, A.P.MAGNUS: Accuracy of simple difference-differential equations for blood flow in arteries. Math. and Comp. in Simulation 21 (1979) 340-351.

• 2 André HAUTOT, A.P.MAGNUS: Calculation of the eigenvalues of Schrödinger equations by an extension of Hill's method, J. Comp. Appl. Math. 5 (1979) 3-15.  pdf file 1300K

• 1 A.P.MAGNUS Fractions continues généralisées et matrices infinies, Bull. Soc. Math. Belg. 29 (ser.B) (1977) 145-159.

A paraître:

En préparation:

V. Pierrard, and A. Magnus, Lorentzian orthogonal polynomials

Marc Germain, Vincent van Steenberghe, Alphonse Magnus, Optimal policy with tradable and bankable pollution permits : Taking the market microstructure into account.

J.Lemaire, V.Pierrard, A.Magnus, Overview of models for parallel electric fields distributions and double layer solutions in magnetospheric physics.

A.Magnus, Freud equations for Legendre polynomials on a circular arc and solution of the Grünbaum-Delsarte-Janssen-Vries problem.

Sélection d'abstracts.

87h:42039 (18:11) 42C05

Magnus, Alphonse P. (B-UCL)

On Freud's equations for exponential weights. (English)

Papers dedicated to the memory of Géza Freud.

Journal of Approximation Theory 46 (1986), no. 1, 65-99.

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 exp(-|x|^a), a > 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 xp_n(x)=a_n+1p_n+1(x)+b_np_n(x)+a _np_n-1(x), then (*)

lim n®¥

an/n1/a

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 a is an even integer; more precisely, the existence of (*) is proved for weights of the form exp(-P(x)) where P is a polynomial of degree 2m, m=1,2,¼, with positive leading coefficient. The technique of the proof is very fine; it starts from an identity of Freud for the a_n's.

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.

See also the following review.

{For the entire collection see MR 87a:41003.}

Reviewed by Totik, V. (Szeged)

Cited in: 88d:42039

Copyright American Mathematical Society 1987, 1995

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88i:65022 (19:16) 65D15

Magnus, Alphonse P. (B-UCL)

Toeplitz matrix techniques and convergence of complex weight Pade approximants. (English)

Journal of Computational and Applied Mathematics 19 (1987), no. 1, 23-38.

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.

Reviewed by Rodrigues, A. J. (Nairobi)

Copyright American Mathematical Society 1988, 1995

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A.P.MAGNUS: On the use of Carathéodory-Fejér method for investigating '1/9' and similar constants, pp. 105-132 in A.CUYT, Editor: Nonlinear Numerical Methods and Rational Approximation , D.Reidel, Dordrecht 1988.

Let E_n be the error norm of the best L_¥ rational approximation of degree n to the exponential function exp(-t) on [0,¥). Grounds are given for setting the conjectured limit E_n/qⁿ® 2q^1/2 when n®¥, where q is the known constant `1/9'= 1/9.2890254919208189187554494359517450610316948677¼, based on the singular values and functions of the relevant Hankel operator (Carathéodory-Fejér's method). Moreover, hints are given according to which a valuable asymptotic expansion of E<sub>n</sub> should also contain n<sup>th</sup> powers of new constants q<sub>1</sub>=`1/56¢, q₂=`1/240¢, etc.

A vrai dire, cet article-partie d'actes est assez peu cité, on cite plutôt

(cf. A.A. GONCHAR, E.A. RAKHMANOV, Equilibrium distribution and the degree of rational approximation of analytic functions, Mat. Sb.  134 (176) (1987) 306-352 = Math. USSR Sbornik 62 (1989) 305-348,

R.S. VARGA, Scientific Computation on Mathematical Problems and Conjectures, CBMS-NSF Reg. Conf. Series in Appl. Math. 60 , SIAM, Philadelphia, 1990)

une lettre que j'ai envoyée à une dizaine de personnes... Les russes, prudents avec les questions ethniques, ont jugé bon de la référencer comme ``Preprint B-1348, Inst. Math., Katholieke Univ. Leuven, Louvain, 1986''.

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95f:33011b (94:16) 33C45 26C05 42C05

Erdélyi, Tamas (3-SFR) ; Magnus, Alphonse P. (B-UCL-PA) ; Nevai, Paul (1-OHS)

``Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials''. (English) SIAM J. Math. An. 25 (1994) 602-614.

Erratum: SIAM Journal on Mathematical Analysis 25 (1994), no. 5, 1461.

It is well known that the Legendre polynomial P_n(x) satisfies the Bernstein inequality

(sinq)1/2 |Pn(cosq)| < (2/p)1/2 n-1/2,    0 £ q £ p.

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 n^-1/2 can be replaced by (n+1/2)^-1/2. For ultraspherical polynomials P_n^(l)(x), L. Lorch [Appl. Anal. 14 (1982/83), no. 3, 237-240; MR 84k:26017] proved that

(sinq)l | Pn(l)(cosq)| < 21-l G(l)-1 (n+l)l-1

for 0 < l < 1 and 0 £ q £ 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 P_n^(a,b)(x) with -1/2 < a, b < 1/2, and a+b > 0.

Here, in this interesting paper, a more general result is obtained. For orthonormal Jacobi polynomials p_n(w,x) with respect to the weight w(x)=(1-x)^a(1+x)^b with a ³ -1/2 and b ³ -1/2, the inequality

max x Î [-1,1]  Ö   1-x2   w(x) pn(w,x)2 £ 2e(2+ Ö   a2+b2   ) p

holds for every n ³ 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 l_n(w,x) and obtain a Riccati equation which yields estimates for p_n(w,x)²l_n(w,x).

Reviewed by Guadalupe Hernández, José Javier (Logrono)

Copyright American Mathematical Society 1995

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A. P. MAGNUS: Freud's equations for orthogonal polynomials as discrete Painlevé equations.

We consider orthogonal polynomials p_n with respect to an exponential weight function w(x)=exp(-P(x)). 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. Freud's contribution).

Most striking example is n=2tw_n+w_n(w_n+1+w_n+w_n-1) for the recurrence coefficients p_n+1=xp_n-w_n p_n-1 of the orthogonal polynomials related to the weight w(x)=exp(-4(tx²+x⁴)). This example appears in practically all the references. The connection with discrete Painlevé equations is described here.

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Activités d'édition.

Je suis membre de l'editorial board du Journal of Approximation Theory.

Collaboration à l'organisation de congrès.

Polynômes orthogonaux et applications, Bar-le-Duc, 15-18 octobre 1984.

Invitations à des conférences et congrès.

Recursion Conference and Applications, Imperial College, London, 13 & 14 septembre 1984.

Third International Symposium on Orthogonal Polynomials and their Applications, Erice (Trapani), 1-8 juin 1990.

Constructive Methods in Complex Analysis, Oberwolfach, 26 mars-1 avril 1995.

Symmetries and Integrability of Difference Equations II, Canterbury, 1-5 juillet 1996.

  Colloque OPSFA "Orthogonal polynomials, Special functions and Applications" 2 au 6 juillet 2007  http://www.cirm.univ-mrs.fr/liste_rencontre/Rencontres2007/Valent07/Valent07.html

3èmes Journées Approximation May 15-16, 2008, Laboratoire Paul Painlevé UMR 8524, Université de Lille 1, FRANCE

Workshop: "Elliptic integrable systems, isomonodromy problems, and hypergeometric functions". Yu.I. Manin, M. Noumi, E.M. Rains, H. Rosengren, V.P. Spiridonov (organizers) Hausdorff Center for Mathematics, Bonn 21-25 July, 2008 http://www.hausdorff-center.uni-bonn.de/elliptic-integrable-systems

Présences à des conférences et congrès.

Padé Approximation and its Applications , Antwerpen 1979.

Padé Approximation and its Applications , Amsterdam 1980.

Padé Approximation and its Applications , Bad Honnef 1983.

Padé Approximation and its Applications , Marseille-Luminy, 14-18 October 1985.

Nonlinear Numerical Methods and Rational Approximation , Antwerpen, 1987.

2^nd International Symposium on Orthogonal Polynomials and their Applications , Segovia 1988.

4^th International Symposium on Orthogonal Polynomials and their Applications , Evian 1992.

Nonlinear Numerical Methods and Rational Approximation II , Antwerpen, 1993.

Computation in Economics, Finance and Engineering: Economic Systems, Cambridge (GB), 1998.

International conference on rational approximation, Anvers, 6-11 juin 1999.