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Title: Ecuaciones en Derivadas Parciales no lineales: difusión, explosión y fronteras libre
Code: MTM2005-08760-C02-02
Center: Ministerio de Ciencia e Innovación de España
Head: Arturo de Pablo Martínez
Period: January, 2005- December, 2008
Number of participants: 4
Summary: In this project we study partial differential equations and their applications to the mechanics of continuous media. The topics covered are theory of nonlinear evolution equations, free boundaries, self-similarity and asymptotic methods, problems in combustion theory, blow-up and extinction, numerical analysis and simulation.
Title: Ortogonalidad y Aproximación. Teoría y Aplicaciones Físicas y Técnicas.
Code: MTM2006-13000-C03-02
Center: Ministerio de Ciencia e Innovación de España.
Head: Guillermo López Lagomasino
Period: October, 2006- September, 2009
Number of participants: 20
Summary: The aim of this project is, on one hand, to investigate the analytic properties of orthogonal polynomials with respect to several models of orthogonality and, on the other, to explore scientific, technological and medical (medical diagnosis by image) application of this study. More specifically, we will study the orthogonality: (a) of matrix type: with respect to a positive definite matrix of measures on the real line; furthermore, we explore the presence of matrix orthogonal polynomials that verify second order differential equatons in modeling quantum relativistic systems (Dirac equation) with coulombian potential, as well as the corresponding time and limiting problems presumably associated with problems of limited angles in tensor tomography; (b) of Sobolev type: where the derivatives of the polynomials appear; Sobolev orthogonal polynomials present advantages for the numerical treatment, by means of spectral methods, of boundary value problems of (ordinary and partial) differential equations, as well as in approximation problems in Fourier-Sobolev series; (c) with respect to varying measures and its application to the study of certain (infinite dimensional SIMO) dynamical systems; (d) q-polynomials and other special functions and its application to the modeling of different discrete oscillatory quantum systems, and other physical and biological systems such as macromolecules and molecular motors. We will also consider other closely related fields: moment problems, rational approximation (mainly Pade approximation and its generalizations, with applications to the study of the stability of time delay dynamical systems, as well as computational methods for special functions relevant in physical and mathematical models), Fourier series and operator theory. The techniques used are, mainly, of matrix analysis, potential theory, Fourier analysis, operator theory, interpolation and classical complex analysis.

List of Publications.

  1. L. D. Abreu, F. Marcellán, S. Yakubovich. Hardy type theorem for orthogonal functions with respect to their zeros. The Jacobi weight case.  J. Math. Anal. Appl. 341 (2008), 803-812.
  2. M. Alfaro, F. Marcellán, A. Peña y M. L. Rezola. When do linear combinations of orthogonal polynomials yield new sequences of orthogonal polynomials?. J. Comput.  Appl. Math. 233 (2010) 1446-1452.
  3. J.  Arvesú. On some properties of q-Hahn multiple orthogonal polynomials.  J.   Comput.   Appl. Math. (2009), 1462-1469.
  4. D. Barrios Rolanía, A. Branquinho, A. Foulquié Moreno. Dynamics and interpretation of some integrable systems via multiple orthogonal polynomials. J.  Math. Anal. Appl. 361 (2010), 358-370
  5. D. Barrios Rolanía, A. Branquinho, Complex high order Toda lattices, J. Difference Equations and Applications. 15 (2009), 197-213
  6. D. Barrios Rolanía, J.R. Gascón Márquez.  Spectrum and Generation of Solutions of the Toda Lattice, Discrete Dynamics in Nature and Society 2009, Article ID 23748 doi:10.1155/2009/237487 (2009)
  7. E. Berriochoa, A. Cachafeiro, F. Marcellan. A new quadrature formule on the unit circle. Numer. Algor. 44 (2007), 391-401.
  8. E. Berriochoa, A. Cachafeiro, J. M. García-Amor, F. Marcellán. New quadrature rules for  Bernstein measures on the interval [-1,1]. Electr. Trans.  Numer. Anal. 30 (2008), 278-290.
  9. A. Branquinho, U. Fidalgo,  A. Foulquié Moreno. Riemann-Hilbert problem associated with Angelesco systems. J. Comp. Appl. Math. 233 (2009), 643-651.
  10. A. Branquinho, A. Foulquié, F. Marcellán, M. N. Rebocho. Coherent pairs of linear functionals on the unit circle. J. Approx. Theory 153 (2008), 122-137.
  11. B. de la Calle Ysern, P. González Vera. Rational quadrature formulae on the unit circle with arbitrary poles. Numer. Math. 107 (2007), 559-587.
  12. B. de la Calle Ysern, F. Peherstorfer,  Ultraspherical Stieltjes polynomials and Gauss-Kronrod quadrature behave nicely for \lambda<0. SIAM J. Numer. Anal. 45 (2007), 770-786.
  13. B. de la Calle Ysern, G. López Lagomasino, L. Reichel. Stieltjes-type polynomials on the unit circle. Math. Comp. 266 (2009), 969-997.
  14. K. Castillo, L. Garza, F. Marcellan. Laurent Polynomials Perturbations of Linear Functionals. An  inverse problem. Electronic Transactions in Numerical Analysis 36 2010, 83-98.
  15. R. S. Costas-Santos. On the elementary symmetric functions of a sum of matrices. J. Algebra Number Theory, Adv. Appl.   1  (2009), 99-112.
  16. R. S. Costas-Santos,  C. R. Johnson, B. Tadchiev. Matrices Totally Positive Relative to a Tree. Electronic Linear Algebra  18  (2009),   211-221.
  17. R. S. Costas-Santos, F. Marcellan. Q-classical orthogonal polynomials. A general approach. Acta Appl. Math. (2010). Doi: 10.1007/ s10440-009-9536-z
  18. R. S. Costas-Santos, J. F. Sánchez-Lara Extensions of discrete classical orthogonal polynomials beyond the orthogonality. J. Comput. Applied Math. 225 (2009), 440-451.
  19. M.A. Delgado, Jeffrey S. Geronimo, Plamen Iliev, Yuan Xu, On a Two-Variable Class of Bernstein–Szego" Measures. Constructive Approx. 30 (2009), 71-91,
  20. J.I. de Vicente, Lower bounds on concurrence and separability conditions. Physical Review A 75, 052320 (2007)
  21. J.I. de Vicente.  Further results on entanglement detection and quantification from the correlation matrix criterion. Journal of Physics A:  41, 065309 (2008).
  22. J.I. de Vicente, S. Gandy, J. Sánchez-Ruiz, Information entropy of Gegenbauer polynomials of integer parameter. Journal of Physics A  40, 8345 (2007).
  23. J.I. de Vicente, J. Sánchez-Ruiz, Improved bounds on entropic uncertainty relations. Physical Review A  77, 042110 (2008).
  24. H. Dueñas, F. Marcellan. Laguerre type orthogonal polynomials. Electrostatic interpretation. Int. J. Pure Appl. Math. 38 (2007), 345-358.
  25. H. Dueñas, F. Marcellán. Jacobi-Type orthogonal polynomials: holonomic equation and electrostatic interpretation. Commun. in the Analytic Theory of Cont. Fractions 15 (2007), 4-19.
  26. H. Dueñas, F. Marcellán. Perturbations  of Laguerre-Hahn functional. Modification by the derivative  of a Dirac delta. Integral   Transf. and Special Funct. 20 (2009), 59-77.
  27. H. Dueñas, F. Marcellan. The Laguerre Sobolev orthogonal polynomials. J. Approx. Theory 162 (2010), 421-440.
  28. H. Dueñas, F. Marcellan. The holonomic equation of the Laguerre-Sobolev type orthogonal polynomials: A non diagonal case. J. Difference Equations Appl. (2010) doi: 10/1080/10236190903456063
  29. H. Dueñas, F. Marcellan. Asymptotic behaviour of Laguerre-Sobolev type orthogonal polynomials. A non diagonal case. J. Comput. Appl. Math. (2010). Doi: 10/1016/j. cam. 2009.07.055.
  30. B. Xh. Fejzullahu, F. Marcellán. On convergence and divergence of Fourier expansions with respect to some Gegenbauer-Sobolev type inner product. Commun. in the Analytic Theory of Continued Fractions 16 (2009), 1-11.
  31. B. Xh. Fejzullahu, F. Marcellán. A Cohen type inequality for Laguerre Sobolev expansions, J. Math. Anal. Appl. 352 (2009), 880-889.
  32. B. Xh Fejzullahu, F. Marcellan. Asymptotic properties of orthogonal polynomials with respect to a non discrete Jacobi-Sobolev inner product. Acta Appl. Math. (2010). Doi: 10.1007/ s10440-009-9511-8.
  33. U. Fidalgo Prieto, J. Illán, G. López Lagomasino. Convergence and computation of simultaneous rational quadrature rules. Numerische Mathematik 106 (2007), 99-128.
  34. U. Fidalgo Prieto, G. López Lagomasino. Generalized Hermite-Padé approximation for Nikishin systems of three functions. J.  Comput. Appl. Math. 233 (2010), 1525-1533.
  35. U. Fidalgo Prieto, A. López García, G. López Lagomasino, V. N. Sorokin. Mixed type multiple orthogonal polynomials for two Nikishin systems. Constructive Approximation   DOI:10.1007/s00365-009-9077-8.
  36. L. Garza, J. Hernandez, F. Marcellán. Orthogonal polynomials and measures on the unit cicle. The Geronimus  Transformations. J. Comput.  Appl. Math. 233 (2010), 1220-1231.
  37. L. Garza, F. Marcellán. Spectral transformations of measures supported on the unit circle and the Szegö transformation. Numer. Algor. 49 (2008), 169-185.
  38. L. Garza, F. Marcellán.  Szegö transforms and Nth order associated polynomials on the unit circle. Computers  and  Math. Appl. 57 (2009), 1659-1671.
  39. L. Garza, F. Marcellán, Linear spectral transformations and Laurent polynomials. Med.  J.  Math. 6 (2009), 273-278.
  40. L. Garza, F. Marcellán. Verblunsky parameters and linear spectral transformations. Meth. Appl. Analysis 16 (2009), 69-86.
  41. L. Garza, F. Marcellán. Szegö transformations and rational spectral transformations for associated polynomials. J. Comput. Appl. Math. 233 (2009), 730-738.
  42. J. Hernandez, F. Marcellán. Geronimus spectral transforms and measures on the complex plane. J. Comput. Appl. Math. 219 (2008), 441-456.
  43. R. H. Heredero, D. Levi, M. Petrera, C. Scimiterna, Multiscale  expansion on the lattice and integrability of partial difference  equations. Journal of Physics A 41 (2008) 1-12.
  44. R. H. Heredero, D. Levi, M. Petrera, C. Scimiterna, Multiscale  expansion and integrability properties of the lattice potential KdV  equation. J. of Nonlinear Math. Phys.15 (2008) 313-323.
  45. R. H. Heredero, E. Reyes. Nonlocal symmetries and a Darboux  transformation for the Camassa–Holm equation. Journal of Physics A, 42 (18) (2009) FTC 182002.
  46. A. López, G. López Lagomasino. Ratio asymptotics of Hermite-Padé orthogonal polynomials mials for Nikishin systems: II. Advances in Mathematics 218 (2008), 1081-1106.
  47. A. López, G. López Lagomasino. Relative asymptotics of multiple orthogonal polynomials of Nikishin systems. J. of Approx. Theory 158 (2009), 214-241
  48. G. López Lagomasino, A. Martínez, I. Pérez, H. Pijeira. Strong asymptotics for Sobolev orthogonal polynomials in the complex plane. J. Math. Anal. Appl. 340 (2008), 521–535.
  49. G. López Lagomasino, J. Mínguez. Fourier-Padé approximants for Nikishin systems. Constructive Approximation 30 (2009), 53-69.
  50. G. López Lagomasino, D. Pestana, J. M. Rodríguez, D. Yakubovich. Computation of conformal representations of compact Riemann surfaces. Math. of Comp. 79 (2010), 365-382.
  51. G. López Lagomasino, L. Reichel, L. Wunderlich. Matrices, moments and rational quadrature. Linear Algebra and its Appl. 429 (2008), 2540-2554.
  52. A. Martínez Finkelshtein,  A.M. Delgado, G.M. Castro, A. Zarzo, J.L. Alió. Comparative análisis of some modal reconstructionmethods of the shape of the cornea from corneal elevation data. IOVS 50 (2009), 5639-5645.
  53. F. Marcellán, R. Sfaxi. Inverse finite-type relations between sequences of polynomials. Rev. Acad. Colombiana de Ciencias Exactas, Fisicas y Naturales 32 (123) (2008), 245-255.
  54. H. Pijeira, C. Díaz, R. Orive. Zeros and logarithmic asymptotics of contracted Sobolev orthogonal polynomials for exponential weights. J. Comp. Appl. Math. 233 (2009),  691-698.
  55. H. Pijeira, C. Díaz, R. Orive. Asymptotically extremal polynomials with respect to varying weights and application to Sobolev orthogonality. J. Math. Anal. Appl. 346 (2008) 480-488.
  56. A. Portilla, Y. Quintana, J. M. Rodríguez, E. Tourís. Weighted Weierstrass' Theorem with first derivatives. J. Math. Anal. Appl. 334 (2007), 1167-1198
  57. A. Portilla, Y. Quintana, J. M. Rodríguez,  E. Tourís.  Weierstrass' Theorem in weighted Sobolev spaces with k derivatives. Rocky Mountain J. of Mathematics 37 (2007), 1989-2024.
  58. A. Portilla, J. M. Rodríguez, E. Tourís, Stability of Gromov hyperbolicity. Journal of Advanced Mathematical Studies 2 (2009), 1-20.
  59. A. Portilla, J. M. Rodríguez, E. Tourís, The multiplication operator, zero location and asymptotic for non-diagonal Sobolev norms, Acta Appl. Math. DOI 10.1007/s10440-009-9541-2.
  60. A. Portilla, E.Tourís. A new characterization of Gromov hyperbolicity of surfaces with negative variable curvature. Publications Matematiques, 53 (2009), 83-10.
  61. J. M. Rodríguez, A simple characterization of weighted Sobolev spaces with bounded multiplication operator, J. Approx. Theory 153 (2008), 53-72.
  62. J. M. Rodríguez, J. M. Sigarreta, Sobolev spaces with respect to measures in curves and zeros of Sobolev orthogonal polynomials, Acta Appl. Math. 104 (2008), 325-353.
Title: Geometric theory of functions
Code: MTM 2009-07800
Center: Ministerio de Ciencia e Innovación de España
Head: José M. Rodríguez
Period: January, 2010- December, 2012
Number of participants: 10
Summary: This project intends to study the close connection between the theory of functions, ergodic theory and geometry of negative curvature, in the context of Riemann surfaces and Riemannian manifolds with variable negative curvature. The problems proposed are continuation of current research by the members of the research team. More precisely, we study: localization and asymptotic behaviour of geodesics of Riemann surfaces and hyperbolic spaces, radial behaviour of holomorphic functions defined in the unit disc, dynamic metric systems, recurrence, asymptotic values of quasi-regular functions, and geometric characterization of certain Riemann surfaces.
Title: Ortogonalidad, Teoría de Aproximación y Aplicaciones.
Code: BFM2003-06335-C03-02
Center: Ministerio de Educación y Ciencia de España.
Head: Guillermo López Lagomasino
Period: November, 2003- Novermber, 2006
Number of participants: 20
Summary: The aim of this project is to investigate the analytical properties of orthogonal polynomials with respect to different orthogonality models: (a) classical: with respect to a positive measure (on the real line, on the unit circle or, in general, on any subset of the complex plane); (b) matrix: with respect to a positive definite matrix of measures on the real line; (c) Sobolev: where the derivatives of polynomials are involved; together with other related fields such as: moment problems, q-polynomials (with applications to different discrete models of cuantic oscilators), rational approximation (Padé type approximants and their extensions, with applications to estability of retarded dynamical systems together with computational methods for special functions which are relevant in physical-mathematical models) and, in general, approximation theory. The techniques used are Matrix Analysis, Potential Theory, Fourier Analysis, Operator Theory and Interpolation, which provide a theoretical frame for the computational aspects involved in such problems.

List of Publications.

  1. U. Fidalgo, G. López Lagomasino. General results on the convergence of multipoint Hermite-Padé approximants of Nikishin systems. Constructive Approximation, 25 (2007), 89-107.
  2. D. Barrios Rolanía, G. López Lagomasino. Asymptotic behavior of solutions of general three term recurrence relations. Advances in Computational Mathematics, 26 (2007): 9-37.
  3. U. Fidalgo, G. López Lagomasino. Rate of convergence of generalized Hermite-Padé approximants of Nikishin systems. Constructive Approximation, 23 (2006), 165-196.
  4. D. Barrios, B. de la Calle, G. López Lagomasino. Ratio and relative asymptotic of polynomials orthogonal with respect to varying Denisov-type measures. J. of Approx. Theory, 139 (2006), 223-256.
  5. A. I. Aptekarev, V. Kalyagin, G. López Lagomasino, I. A. Rocha. On the limit behavior of the recurrence coefficients for multiple orthogonal polynomials. J. of Approx. Theory, 139 (2006), 346-370.
  6. M. Bello, A. Martínez, X. Tolsa, G. López Lagomasino. Sesión monográfica de teoría de aproximación. Boletín SEMA, 34 (2006), 153-174.
  7. G. López Lagomasino, F. Marcellán, H. Pijeira. Logarithmic asymptotic of contracted Sobolev extremal polynomials on the real line. J. of Approx. Theory, 143 (2006), 62-73.
  8. A. I. Aptekarev, G. López Lagomasino, I. A. Rocha. Ratio asymptotic of Hermite-Padé polynomials for Nikishin systems. Matematicheskii Sbornik, 196 (2005), 1089-1107.
  9. G. López Lagomasino, I. Pérez, H. Pijeira. Asymptotic of extremal polynomials in the complex plane. J. of Approx. Theory, 137 (2005), 226-237.
  10. M. Alfaro, F. Marcellán, A. Peña, M. L. Rezola, On linearly related orthogonal polynomials and their functionals, J. Math. Anal. and Appl. 287 (2003) 307-319.
  11. J. Hernández, F. Marcellán, C. Rodríguez. Leverrier-Fadeev algorithm and classical orthogonal polynomials, Rev. de la Acad. Col. de Cien. Exactas, Fís. y Nat.  28 (106) (2004) 39-47.
  12. M. I. Bueno, F. Marcellán,  Darboux Transformation and Perturbation of  Linear Functionals, Linear Alg. and Appl. 384 (2004) 215-242.
  13. R. Alvarez-Nodarse, J. Arvesu, F. Marcellán. On the Krall-type polynomials, J. of Appl. Math. 5 (2004) 359-370.
  14. J. Arvesu, A. Garrido, F. Marcellán, Modification of linear functionals with Dirac masses: Class of the modified linear functional, Bol. Mat.  N.S. 11 (1) (2004) 32-51.
  15. M. Alfaro, F. Marcellán, A. Peña, M. L. Rezola. On rational transformations of linear functionals. A direct problem, J. Math. Anal. and Appl. 298 (2004) 171-183.
  16. J. Hernández, F. Marcellán. An extension of Leverrier-Faddeev algorithm using a basis of classical orthogonal polynomials,  Facta Univ.  Ser. Math. Inform. 19 (2004) 73-92.
  17. A. M. Delgado, F. Marcellán. Companion linear functionals and Sobolev inner products, Meth. and   Appl. of Anal. 11 (2004) 237-266.
  18. U. Fidalgo, J. Illán, G. López Lagomasino. Hermite Padé approximation and simultaneous quadrature formulas. J. of Approx. Theory   126 (2004), 171-197.
  19. A. Gil, J. Segura, N.M. Temme. Computing Solutions of the Modified Bessel Differential Equation for Imaginary Orders and Positive Arguments. ACM Trans. Math. Softw. 30 (2004), 145-158.
  20. A. Gil, J. Segura, N.M. Temme.  Algorithm 831: Modified Bessel Functions of Imaginary Orders and Positive Argument. ACM Trans. Math. Softw. 30 (2004), 159-164
  21. A. Deaño, A. Gil, J. Segura. New inequalities from classical Sturm theorems. J. Approx. Theory 131 (2004), 208-230.
  22. G. López Lagomasino, I. Pérez, H. Pijeira. Asymptotic of extremal polynomials in the complex plane. J. of Approx. Theory 137 (2005), 226-237.
  23. I. Area, E. Godoy, F. Marcellán,  J. J. Moreno-Balcázar. Asymptotics of Sobolev orthogonal polynomials for ?-coherent pairs of  Meixner type.  J. of Comput. and Appl. Math. 178 (2005) 21-36.
  24. A. M. Delgado, F. Marcellán, On an extension of symmetric coherent pairs of orthogonal polynomials, J. of Comput. and Appl. Math.  178 (2005) 155-168.
  25. J.S. Geronimo, D.S. Lubinsky, F. Marcellán. Asymptotics for Sobolev Orthogonal Polynomials for Exponential Weights, Constr. Approx. 22 (2005) 309-346.
  26. M. Isabel Bueno, F. Marcellán, J. Sánchez Ruiz. Continuous symmetrized Sobolev inner products of order N (I), J. Math. Anal. and Appl.  306 (2005) 83-96.
  27. J. Arvesu, A. Garrido, F. Marcellán. An electrostatic interpretation of the zeros of the Freud type orthogonal polynomials,  Electr. Trans. in Num. Anal. 19 (2005) 37-47.
  28. J. Hernández , F. Marcellán, Christoffel transforms and Hermitian linear functionals, Med. J. of Math. 2 (2005) 451-458.
  29. José M. Rodríguez, Dmitry V. Yakubovich. A Kolmogorov-Szego-Krein type condition for weighted Sobolev spaces. Indiana University Mathematical Journal 54 (2005), 575-598.
  30. V.I. Kolyada. Inequalities of Gagliardop-Niremberg type and estimates for the moduli of continuity. Russian Math. Surveys 60 (2005), 1147-1164.
  31. B. de la Calle. Error bounds for rational quadrature formulas of analytic functions.  Numerische Mathematik 101 (2005), 251-271.
  32. I. Alvarez Rocha, L. Salto. "Asymptotics of polynomials orthogonal with respect to a discrete-complex Sobolev inner product", J. Comp. and Appl. Math. 178 (2005), 1-19.
  33. J. I. de Vicente, J. Sánchez Ruiz. Separability conditions from the Landau-Pollak uncertainty relation. Physical Review A  71, 052325 (2005)
  34. J. Sánchez Ruiz, J. S. Dehesa. Fisher information of orthogonal hypergeometric polynomials. J.  of Comput.  and Appl. Math.  182 (2005), 150-164.
  35. J. Arvesú. Quantum algebras suq(2) and suq(1,1) associated with certain q-Hahn polynomials: A revisited approach. Electr. Trans. Num. Anal. 24 (2006), 24-44.
  36. M. I. Bueno, F. Marcellán. Polynomials perturbations of bilinear functionals and Hessenberg matrices. L. Alg. and Appl. 414 (2006), 64-83.
  37. A. M. Delgado, J. S. Geronimo, P. Iliev , F. Marcellán. Two variable orthogonal polynomials and structured matrices,   SIAM J. Matrix Anal. and Appl.  28 (2006) 118-147.
  38. G. Lopez, F. Marcellán, H. Pijeira. Logarithmic Asymptotics of contracted Sobolev extremal polynomials, J. Approx. Theory  143 2006), 62-73.
  39. M. Isabel Bueno, F. Marcellán, J. Sánchez Ruiz. Continuous symmetrized Sobolev inner products of order N (II), Electr. Trans. in Num. Anal.  24 (2006) 55-65.
  40. F. Marcellán, S. M. Zagorodnyuk. On the basic set of solutions of a high order linear difference equation. J. of Diff. Eq. and Appl. 12 (2006) 213-228.
  41. A. I. Aptekarev, A. Cachafeiro, F. Marcellán. A scalar Riemann boundary value problem approach to the orthogonal polynomials on the circle, J. Approx. Theory 141 (2006) 174-181.
  42. V.I. Kolyada, Mixed norms and Sobolev type inequalities. Banach Center Publ. (2006),  v. 72, 141-160.
  43. J. Hernández, F. Marcellán. Classical orthogonal polynomials and Leverrier-Faddev algorithm for the matrix pencils sE – A,  Intern. J. of  Math. and Math. Sci. (2006). Article ID 74507.
  44. José M. Rodríguez, Venancio Alvarez, Elena Romera, Domingo Pestana. Generalized weighted Sobolev spaces and applications to Sobolev orthogonal polynomials: a survey. Electronic Transactions in Numerical Analysis 24 (2006), 88-93.
  45. Ana Portilla, Yamilet Quintana, José M. Rodríguez, Eva Tourís. Weierstrass' Theorem in weighted Sobolev spaces with k derivatives: announcement of results. Electronic Transactions in Numerical Analysis 24 (2006), 103-107.
  46. U. Fidalgo,  G. López Lagomasino. Rate of convergence of generalized Hermite-Padé approximants of Nikishin systems. Constructive Approximation  23 (2006), 165-196.
  47. D. Barrios, B. de la Calle, G. López Lagomasino. Ratio and relative asymptotic of polynomials orthogonal with respect to varying Denisov-type measures.   J. of Approx. Theory 139 (2006), 223-256.
  48. A. I. Aptekarev, V. Kalyagin, G. López Lagomasino, I. A. Rocha. On the limit behavior of the recurrence coefficients for multiple orthogonal polynomials  J. of Approx. Theory  139 (2006), 346-370.
  49. A. Gil, J. Segura y N.M. Temme. The ABC of Hyper Recursion. J. Comput. Appl. Math. 190 (2006), 270-286.
  50. A. Gil, J. Segura y N.M. Temme. Computing the Real Parabolic Cylinder Functions U(a,x), V(a,x). ACM Trans. Math. Softw. 32(1) (2006), 70-101.
  51. A. Gil, J. Segura y N.M. Temme. Algorithm 850:Real Parabolic Cylinder Functions U(a,x), V(a,x). ACM Trans. Math. Softw. 32(1) (2006), 102-112.
  52. R. Álvarez-Nodarse, Y. F. Smirnov, R. S. Costas-Santos. A q-analog of the Racah polynomials and q-algebra SUq(2) in quantum optics.  J. of Russian Laser Research, 27 (2006), 1-32.
  53. R. Álvarez-Nodarse, R.S. Costas-Santos. Limit relations between q-Krall type orthogonal polynomials. J. Math. Anal. Appl. 322 (2006), 158–176.
  54. J. L. Fernández, M. V. Melián, D. Pestana. Quantitative mixing results and inner functions. Math. Annalen 337 (2007), 233-251.
  55. F. Marcellán, Non standard orthogonal polynomials. Applications in Numerical Analysis and Approximation Theory, Rev. de la Acad. Col. de Cien. Exactas, Fís. y Nat. 29 (118) (2007).
  56. M. Bello, G. López Lagomasino, J. Míngues. Fourier-Padé approximation for Angelesco systems. Constructive Approximation (on-line, diciembre, 2006)
  57. A. Cachafeiro, F. Marcellán, C. Pérez, Orthogonal Polynomials with respect to the sum of an arbitrary measure and a Bernstein Szegö measure, Adv. in Comput. Math. (on-line, agosto 2006).
  58. L. Daruis, J. Hernandez, F. Marcellán, Spectral Transforms for Hermitian Toeplitz matrices, J. of  Comput. and Appl. Math. En prensa.
  59. F. Marcellán, R. Sfaxi, Second structure relation for semiclassical orthogonal polynomials, J. of  Comput. and Appl. Math. En prensa.
  60. F. Marcellán, A. Martínez Finkelshtein, P. Martínez González, Electrostatic models for zeros of polynomials: old, new, and some open problems,  J. of Comput. and Appl. Math. En prensa.
  61. R.S. Costas, F. Marcellán, Second structure relation for q-semiclassical polynomials of the Hahn Tableau, J. Math. Anal. and Appl. En prensa.
  62. F. Marcellán, J. J. Moreno-Balcazar, Asymptotics and zeros of Sobolev orthogonal polynomials on unbounded supports., Acta Appl. Math. En prensa.
  63. Maria I. Bueno Cachadina, Froilan M. Dopico, A more accurate algorithm for computing the Christoffel transformation, J. of Comp. and Appl. Math. En prensa .
Title: Aproximación Hermite-Padé, ortogonalidad no estándar y aplicaciones.
Code: CCG07-UC3M/ESP-3339
Center: Comunidad de Madrid - Universidad Carlos III de Madrid
Head: Jorge Arvesú Carvallo
Period: January, 2008- December, 2008
Number of participants: 15
Summary: El proyecto abordado trata, por un lado, la investigación de propiedades analíticas y algebraicas de polinomios ortogonales respecto a varios modelos de ortogonalidad y, por otro, la exploración de aplicaciones científicas y tecnológicas, con especial énfasis en aplicaciones físicas y geométricas. Más concretamente, se estudia la ortogonalidad: (a) múltiple: ésta aparece como resultado de la aproximación simultánea a un vector de funciones; (b) Sobolev: donde comparecen las derivadas de los polinomios; los polinomios ortogonales de Sobolev presentan ventajas para el tratamiento numérico mediante métodos espectrales de problemas de frontera para ecuaciones diferenciales (tanto ordinarias como en derivadas parciales), así como en problemas de aproximación en series de Fourier-Sobolev; (c) respecto a medidas variantes y q-medidas. Entre las aplicaciones destaca el estudio de ciertos sistemas dinámicos (SIMO de dimensión infinita); q-polinomios multiortogonales como funciones especiales; el estudio de la hiperbolicidad en diversos dominios (Denjoy) y superficies de Riemann. También se considerarán otras aplicaciones en campos muy relacionados: Problemas de momentos, aproximación racional (principalmente aproximantes de Padé y sus extensiones, con aplicaciones al estudio de la estabilidad de sistemas dinámicos con retardo, junto con métodos computacionales para funciones especiales relevantes en modelos físico-matemáticos), teoría de números, fórmulas de cuadratura, series de Fourier y teoría de operadores. Las técnicas utilizadas son, fundamentalmente, de análisis matricial, teoría del potencial, análisis de Fourier, teoría de operadores, interpolación y análisis complejo clásico. This project deals with the analytic properties of families of orthogonal polynomials with respect to several models of orthogonality and, on the other hand, explores their scientific and technological applications (the modelling of several quantum physical problems and geometrical applications). More precisely, we will focus our attention on three cases of orthogonality: multiple orthogonality, which appears when the orthogonality conditions are considered with respect to vector measures; (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 variant measures and q-measures. Among the applications are considered the study of some dynamical systems (infinite dimensional SIMO systems) and q-multiple orthogonal polynomials as special functions as well as the study of hyperbolicity in domains (Denjoy) and Riemann surfaces. Also other related fields are considered: 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.

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