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Ortogonalidad, Teoría de la Aproximación y sus Aplicaciones en Ciencia y Tecnología. PDF Print E-mail
Written by Guillermo López Lagomasino   
Monday, 08 February 2010 16:48

Title: Orthogonality, Approximation Theory, and its Applications in Science and Technology. 

Code: MTM2009-12740-C03-01.

Center: Ministerio de Ciencia e Innovación de España.

Period: January, 2010- December, 2012.

Head: F. Marcellán Español

Number of participants: 17


Summary 


In this project we are dealing with the analytic properties of families of orthogonal polynomials with respect to several models of inner products and, on the other hand, we explore their scientific and technological applications (the modellisation of several discrete systems of quantum oscillators and other physical and biological systems like macromolecules and molecular motors amoing other illustrative examples). More precisely, we will focus our attention on three cases of orthogonality where the teams involved in the project have a recognized experience and worldwide leadership. (a) Matrix orthogonality with respect to a positive definite matrix of measures supported on the realk line. Here we will deal with the spectral study of second order linear differential operators whose coefficients are matrix polynomials and their eigenfunctions are matrix orthogonal polynomials. As applications, we will consider the modellisation of relativistic quantum systems (Dirac equation) with Coulombian potential, discrete Markov chains when the interactions are not reduced to the closest neighbors and other problems with potential impact on the diagnosis by medical imaging using tensor tomography. (b) Sobolev orthogonality where the derivatives of polynomials are involved in the weighted inner product. These orthogonal polynomials present some advantages with respect to the standard ones when spectral methods are considered in the numerical analysis of boundary value problems both for differential and partial differential equations as well as they improve the standard techniques in Approximation Theory when Fourier-Sobolev expansions are considered. (c) Orthogonality with respect to vector measures and their applications in the study of some dynamical systems (infinite dimensional SIMO systems).

We will also deal with other related fields: Moment problem theory, rational approximation (mainly Padé approximants and their extensions, with applications in the study of the stability of time delay dynamical systems) as well as computational methods for Special Functions of relevance in physical-mathemtical models, Number Theory, numerical quadrature, Fourier series, and Operator Theory. The techniques that we will use are Matrix Analysis, Potential Theory, Fourier Analysis, Operator Theory, Interpolation , and classical Complex Analysis.

 


 

List of Publications

 (you can download a copy of a given paper visiting the home page of a team member)


 1.- F. Marcellán, Xh. Fejzullahu, A Cohen inequality for Fourier expansions of orthogonal polynomials with a non-discrete Jacobi-Sobolev inner product. Journal of  Inequalities and Applications. Volume 2010. Article ID 128746. 22 páginas.

2.- K. Castillo, L. Garza, F. Marcellán, Linear spectral transformations, Hessenberg matrices and orthogonal polynomials. Rendiconti Circolo Matematico di Palermo Serie II Supl. 82 (2010) 3-26.

3.- F. Marcellán, R. Sfaxi, Orthogonal polynomials and second order pseudo-spectral linear differential equations. Integral Transforms and Special Functions 21 (2010) 487-501.

4.- F. Marcellán, F. R. Rafaeli, A note on monotonicity of zeros of generalized Hermite-Sobolev type orthogonal polynomials. Integral Transforms and Special Functions 21 (2010) 831-838.

5.- A. Branquinho, F. Marcellán, A. I. Mendes, Vector interpretation of the matrix orthogonality on the real line. Acta Applicandae Mathematicae 112 ( 2010) 357-383.

6.- Carlos Álvarez-Fernández, U. Fidalgo, Manuel Mañas. The multicomponent 2D Toda hierarchy: generalized matrix orthogonal polynomials, multiple orthogonal polynomials and  Riemann-Hilbert problems.  Inverse Problems 26 (2010) 055009. 15 Páginas.

7.- F. Lledó, E. Vasselli, On the nuclearity of certain Cuntz-Pimsner algebras, Mathematische Nachrichten. 283 (2010) 752 - 757.

8.- J. Arvesú, On some properties of q-Hahn multiple orthogonal polynomials, Journal of  Computational and Applied Mathematics,  233  (2010) 1462-1469.

9.- G. López Lagomasino, I. A. Rocha.  A canonical family of multiple orthogonal polynomials.  Journal of Mathematical Analysis and Applications  372 (2010), 390--401.

10.- A. Portilla, Y. Quintana, J. M. Rodríguez, E. Tourís, Zero location and asymptotic behavior for extremal polynomials   with non-diagonal Sobolev norms, Journal Approximation Theory 162 (2010), 2225-2242.

11.- P. Hästö, A. Portilla, J. M. Rodríguez, E. Tourís, Gromov hyperbolic equivalence of the hyperbolic and   quasihyperbolic metrics in Denjoy domains, Bulletin of the London Mathematical Society 42 (2010), 282-294.

12.- P. Hästö, A. Portilla, J.M. Rodríguez, E. Tourís, Comparative Gromov hyperbolicity results for the hyperbolic and quasihyperbolic metrics, Complex Variables and Elliptic Equations 55 (2010), 127-135.

13.-A. Portilla, J.M. Rodríguez, E. Tourís, The multiplication operator, zero location and asymptotic for non-diagonal Sobolev norms, Acta Applicandae Mathematicae 111 (2010), 205-218.

14.-  E. Huertas, F. Marcellán, On Laguerre-type orthogonal polynomials, Volumen en honor del Profesor  Antonio Martinón con ocasión de su 60 cumpleaños. Universidad de La Laguna. 2010. 267-283.

15.- A. Portilla, J. M. Rodríguez, E. Tourís, A real variable characterization of Gromov hyperbolicity of   flute surfaces, Osaka Journal of  Mathematics 48 (2011), 179-207.

16.- F. Lledó, Modular Theory by example, Contemporary Mathematics 534, American Mathematical Society, Providence, Rhode Island, 2011.73-96.

17.- H. Dueñas, E. Huertas, F. Marcellán, Analytic properties of Laguerre-type orthogonal polynomials. Integral Transforms and Special Functions 22 (2011) 107-122.

18.- M. Alfaro, F. Marcellán, A. Peña, M. L. Rezola, Orthogonal polynomials associated with an inverse quadratic spectral transform. Computers and Mathematics with Applications  61 (2011) 888-900. ( C-3)

19.- K. Castillo, L. Garza, F. Marcellán,  Perturbations on the subdiagonals of Toeplitz matrices. Linear Algebra and its Applications  434 (2011) 1563-1579.

20.- F. Marcellán, F. R. Rafaeli, Monotonicity and asymptotic of zeros of Laguerre-Sobolev-type orthogonal polynomials of higher derivatives. Proceedings of the American Mathematical Society 139 (2011)  3929-3936.

  21.-K. Castillo, L. Garza, F. Marcellan, A new linear spectral transformation assocated with derivatives of Dirac   lilinear functionals. Journal of Approximation Theory 163 (2011) 1834-1853.

22.- F. Marcellán, S. M. Zagorodnyuk, Density of polynomials in some L2 spaces on radial rays in the complex plane. Linear  Algebra and its Applications 435 (2011) 128-146.

23.- F. Marcellán, R. Sfaxi, Lowering operators associated with D-Laguerre-Hahn polynomials. Integral Transforms and Special Functions 22 (2011) 879-893.

24.- B. Xh. Fejzullahu, F. Marcellán, Jacobi-Sobolev orthogonal polynomials: Asymptotcs for N-coherence measures. Journal of Inequalities and Applications  Volume 2011. Article ID 294134. 19 paginas.

25.- R. S. Costas Santos,  J. F. Sánchez Lara,  Orthogonality of q- polynomials for non standard parameters, Journal of Approximation Theory 163 (2011)  1246-1268.


26.- U. Fidalgo, G. López Lagomasino. Nikishin systems are perfect. Constructive Approximation  34 (2011), 297-356.


27.- U. Fidalgo, G. López Lagomasino. Nikishin systems are perfect. Case of unbounded and touching supports. Journal of Approximation Theory 163 (2011), 779-811.


28.- C. Álvarez-Fernández, U. Fidalgo, M. Mañas. Multiple orthogonal polynomials of mixed type: Gauss-Borel factorization and the multi-component 2D Toda hierarchy. Advances in Mathematics,  227 (2011),  1451-1525.


29.- P. Ara, F. Lledó, F. Perera, Basic definitions and results for operator algebras, Contemporary Mathematics 534, American Mathematical Society, Providence, Rhode Island 2011. 157-167.


30.- P. Ara, F. Lledó, F. Perera Editores, Aspects of Operator Algebras and Applications, Contemporary Mathematics Vol. 534, American Mathematical Society, Providence, Rhode Island,  2011.


31.- E. Tourís, Graphs and Gromov hyperbolicity for non-constant negatively  curved surfaces,  Journal of Mathematical Analysis and Applications 380  (2011), 865-881


32.- S. Bermudo, J. M. Rodríguez, J. M. Sigarreta, E. Tourís, Hyperbolicity  and complement of graphs, Applied Mathematical Letters 24 (2011),  1882-1887

33.- J. Bello, H. Pijeira, C. Márquez, W. Urbina, Sobolev-Gegenbauer type orthogonality and a hydrodynamical  interpretation, Integral Transforms and Special Functions 22 (2011), 711-722.

34.-A. Portilla, J. M. Rodríguez, E. Tourís, A very simple characterization of Gromov hyperbolicity for a   special kind of  Denjoy domains, Journal Korean Mathematical Society 48 (2011) 565-583.

35.- P. Hästö, A. Portilla, J. M. Rodríguez, E. Tourís, Uniformly separated sets and Gromov hyperbolicity of domains   with the  quasihyperbolicity metric, Mediterranean Journal of Mathematics  8 (2011), 47- 65.

36.- F. Lledó, O. Post, Spectral approximation for discrete and metric graphs: summary of results, Actas Colloquium 2011. Publicaciones del Departamento de Análisis Matemático. Universidad Complutense de Madrid . Sección 1 Núm. 77,  69-74.

37.-  H. Pijeira, Y.  Quintana, J. M. Rodríguez,  Sobolev formal orthogonality on algebraic curves and extensions of Favard theorem, Jaen Journal of  Approximation 3  (2) (2011), 193-207.

38.- R. Costas-Santos, J. Moreno-Balcázar,  The Semiclassical Sobolev orthogonal polynomials: A general approach. Journal of Approximation Theory 163 (2011),  65-83.

39.- B. Bouras, F. Marcellán, Quadratic decomposition of a Laguerre-Hahn polynomial sequence II. Mediterranean  Journal of  Mathematics 9 (2012)  23-49.

40.- A. Alaya, B. Bouras, F. Marcellán, A non-symmetric second degree semi-classical form of class one,  Integral Transforms and Special Functions. 23 (2012) 149-159.

41.- E. Huertas, F. Marcellán,  F. R. Rafaeli, Zeros of orthogonal polynomials generated by canonical perturbations of measures, Applied Mathematics and Computation 218 (2012) 7109-7127.

42.- W. Chammam, F. Marcellán,  R. Sfaxi, Orthogonal polynomials, Catalan numbers, and a general Hankel determinant evaluation. Linear Algebra and Applications 436 (2012) 2105-2116.

43.- K. Castillo, A new approach to relative asymptotic behavior for discrete-Sobolev orthogonal polynomials on the unit circle, Applied Mathematical Letters 25 (2012) 1000-1004.

44.- U. Fidalgo, P. Pedregal,  A general lower bound for the relaxation of an optimal design problem with a general quadratic cost functional, and a general linear state equation. Journal of Convex Analysis 19 (2012) 281-294.

45.- H. Dueñas,  E. Huertas, F. Marcellán, Asymptotic properties of Laguerre-Sobolev type orthogonal polynomials. Numerical Algorithms 60 (2012) 51-73.

46.- K. Castillo, R. L. Lamblem, A. Sri Ranga, On a moment problem associated with Chebyshev polynomials, Applied Mathematics and Computation 218 (2012), 9571-9574.

47.- F. Marcellán, R. Xh. Zejnullahu, B. Xh. Fejzullahu, E. Huertas, On orthogonal polynomials with respect to certain discrete Sobolev inner product, Pacific Journal of Mathematics  257 (2012), 167-188.

48.- F. Marcellán, N. Pinzón, (1,1) q-coherent pairs, Numerical Algorithms  60 (2012) 223-239.

49.- F. Marcellán, M. Sghaier, M. Zaatra,  On semiclassical linear functionals of class s=2. Classification and integral representations. Journal of Difference Equations and Applications  18 (2012) 973-1000.

50.- A. Branquinho, A. Mendes, F. Marcellán, Relative asymptotics for orthogonal matrix polynomials, Linear Algebra and Applications  437 (2012) 1458-1481.

51.- K. Castillo, L. Garza, F. Marcellán,  Zeros of Sobolev orthogonal polynomials on the unit circle, Numerical Algorithms 60 (2012) 669-681.

52.- J. Arvesú, C. Esposito, A high order q-difference equation for q-Hahn multiple orthogonal polynomials, Journal Difference Equations and Applications 18 (2012) 833-847.

53.- J. Borrego, M. Castro, A. Durán, Orthogonal matrix Polynomials satisfying Differential Equations with Recurrence Coefficients having Non-Scalar Limits. Integral Transforms and Special Functions 23 (2012) 685-700. (C-2)

54.- P. Hästö, H. Linden, A. Portilla, J. M.   Rodríguez, E.  Tourís, Gromov hyperbolicity of Denjoy domains with hyperbolic and  quasihyperbolic metrics, Journal of the Mathematical Society of Japan. 64 (2012) 245-259.

55.- M. J. Atia, M. Benabdallah, R. S. Costas, Zeros of polynomials orthogonal with respect to a signed weight, Indagationes Mathematicae, New Series 23 (2012) 26-31.

56.- F. Marcellán, N. Pinzón, Higher order coherent pairs, Acta Applicanda Mathematicae 121 (2012) 105-135.

57.- K. Castillo, M. V. Mello, F. R. Rafaeli, Monotonicity and asymptotics of zeros of Sobolev type orthogonal polynomials: A general case. Applied Numerical Mathematics  62 (2012) 1663-1671.

58.- A. Portilla, J.M. Rodríguez, E. Tourís, Structure Theorem for Riemannian surfaces with arbitrary curvature, Mathematische Zeitschrift  271 (2012), 45-62.

59.- R. S. Costas, F. Marcellán, The complementary polynomials and the Rodrigues operator of classical orthogonal polynomials, Proceedings American Mathematical Society 140 (2012) 3485-3493.

60.- K. Castillo, R. L. Lamblém, F. R. Rafaeli, A. Sri Ranga, Szegö and para-orthogonal polynomials on the real line. Zeros and canonical spectral transformations. Mathematics of Computation  81 (2012) 2229-2249.

61.- J. Cacoq, G. López Lagomasino. Convergence of row sequences of simultaneous Fourier-Padé  approximation, Jaen J. Approx. 4 (2012), 101-120.

62.- P. Hästö, A. Portilla, J. M. Rodríguez, E. Tourís, Gromov hyperbolicity of Denjoy domains through fundamental domains, Publicationes Mathematicae Debrecen 80 (2012), 295-310.

63.- B. Bouras, F. Marcellán, Quadratic decomposition of a family of Hq -semiclassical orthogonal polynomial sequences. Journal of Difference Equations and Applications 18 (2012) 2039-2057.

64.- U. Fidalgo, S. Medina Peralta, J. Mínguez Ceniceros. Mixed type multiple orthogonal polynomials: Perfectness and interlacing properties. Linear Algebra and Applications 438 (2013) 1229-1239. (C-3)

65.-  S. Delvaux, A. López, and G. López Lagomasino. A family of Nikishin systems with periodic recurrence coefficients. Sbornik Mathemathics 294 (2013) 47-78.

66.- K. Castillo, L. E. Garza, F. Marcellán,  Asymptotic behavior and zeros of Sobolev orthogonal polynomials on the unit circle. Integral Transforms and Special Functions. 24 (2013) 23-38.

67.- F. Marcellán, M. Sghaier, M. Zaatra, Semiclassical linear functionals of class three. The symmetric case.  Journal of Difference Equations and Applications 19 (2013) 162-178.

68.- B. Xh. Fejzullahu,  F. Marcellán, A Cohen  type inequality  for Gegenbauer-Sobolev expansions. Rocky Mountain Journal of Mathematics 43 (2013) 135-148.1-14.

69.- F. Marcellán, Y. Quintana, A. Urieles, On W1,p-convergence of Fourier Sobolev expansions, Journal of Mathematical Analysis and Applications, 398 (2013), 594-599

70.- F. Lledó, On spectral approximation, Foelner sequences and crossed products, Journal of Approximation Theory 170 (2013) 155-171.

71.- B. Xh. Fejzullahu, F. Marcellán,  J. J. Moreno-Balcázar, Jacobi-Sobolev orthogonal polynomials: asymptotics and a Cohen type inequality. Journal of Approximation Theory 170 (2013) 78-93.

72.- J. Cacoq, , B. de la Calle Ysern, G. López Lagomasino, Incomplete Padé approximation and convergence of row sequences of Hermite-Padé approximants. Journal of Approximation Theory 170 (2013), 59-77.

73.- J. Arvesu, A. Soria, First order non homogeneous q-difference equation for Stieltjes function characterizing q-orthogonal polynomials. Journal of Difference Equations and Applications 19  (2013)  814-838.

74.- B. Alaoui, F. Marcellán, R. Sfaxi,  Classical orthogonal polynomials with respect to a lowering operator generalizing the Laguerre operator, Integral Transforms and Special Functions 24 (2013) 636-648.

75.- E. Huertas, F. Marcellán, M. Masjed-Jamei,  A Finite class of orthogonal functions generated by Routh-Romanovski polynomials. Complex Variables and Elliptic Equations. (2013) doi:10.1080/17476933.2012.727406

76.- J. Cacoq, B. de la Calle Ysern, G. López Lagomasino. Direct and inverse results on row sequences of Hermite-Padé approximation.  Constructive Approximation 38 (2013), 133-160.

77.- K. Castillo, F. Marcellán, Generators of rational spectral transformations for nontrivial C-functions, Mathematics of Computation 82 (2013) 1057-1068.

78.- 6.- J. Borrego, H. Pijeira Cabrera, Orthogonality with respect to a Jacobi differential
operator and applications, Journal of Mathematical Analysis and Applications 414 (2013) 491-500.

79.- J. Borrego, On orthogonal polynomials with respect to  a class of differential operators, Applied Mathematics and Computation Appl. Math. Comput. 219  (2013) 7853-7871.

80.- F. Lledó   D. Yakubovich, Foelner sequences and finite operators. Journal Mathematical Analysis and Applications
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81.- A. Portilla, Y. Quintana, J. M. Rodríguez, E. Touris, Concerning asymptotic behavior for extremal polynomials associated to non-diagonal Sobolev norms. J. Funct. Spaces Appl. 2013, Art. ID 628031, 11 pp.K. Castillo, F. Marcellán, Generators of rational spectral transformations for nontrivial C-functions, Mathematics of Computation.

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