Artículos (Análisis Matemático)
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Artículo The fixed point for a transformation of Hausdorff moment sequences and iteration of a rational function(Matematisk Inst., 2008-09-01) Berg, Christian; Durán Guardeño, Antonio José; Análisis Matemático; FQM262: Teoría de la AproximaciónWe study the fixed point for a non-linear transformation in the set of Hausdorff moment sequences, defined by the formula: T((an))n=1/(a0+⋯+an). We determine the corresponding measure μ, which has an increasing and convex density on ]0,1[, and we study some analytic functions related to it. The Mellin transform F of μ extends to a meromorphic function in the whole complex plane. It can be characterized in analogy with the Gamma function as the unique log-convex function on ]−1,∞[ satisfying F(0)=1 and the functional equation 1/F(s)=1/F(s+1)−F(s+1), s>−1.Artículo The algebras of difference operators associated to Krall-Charlier orthogonal polynomials(Elsevier, 2018-06-25) Durán Guardeño, Antonio José; Análisis Matemático; FQM262: Teoría de la AproximaciónArtículo The algebra of difference operators associated to a family of orthogonal polynomials(Elsevier, 2012-01-27) Durán Guardeño, Antonio José; Análisis Matemático; FQM262: Teoría de la AproximaciónGiven a weight matrix W of arbitrary size N X N on the real line and a sequence of matrix valued orthogonal polynomials (Pn)n with respect to W, we study the algebra D(W) of difference operators D with matrix polynomial coefficients such that D (Pn) = TnPn, with Tn E CNXN. As a consequence, we deduce that scalar polynomials orthogonal with respect to a nondegenerate positive measure can satisfy only difference equations of even order, and prove that the algebra of difference operators associated to any of the four discrete classical families of Charlier, Meixner, Krawtchouk and Hahn is generated from the second order difference operator (it is unique up to constants). We also introduce three illustrative matrix examples showing that the situation in the matrix valued case is much more interesting. These matrix families are the first non-trivial examples of weight matrices appearing in the literature whose orthogonal polynomials satisfy second order difference equations.Artículo Symmetries for Casorati determinants of classical discrete orthogonal polynomials(AMS, 2013-11-21) Durán Guardeño, Antonio José; Análisis Matemático; FQM262: Teoría de la AproximaciónGiven a classical discrete family (Pn)n of orthogonal polynomials (Charlier, Meixner, Krawtchouk or Hahn) and the set of numbers m+i-1, k,m ≥ 0 and , we consider the k X k Casorati determinants det((Pn+j-1(m+i-1))kij=1,n 0. In this paper, we conjecture a nice symmetry for these Casorati determinants and prove it for the cases k 0,m = 0,1 and m 0,k = 0,1. This symmetry is related to the existence of higher order difference equations for the orthogonal polynomials with respect to certain Christoffel transforms of the classical discrete measures. Other symmetry will be conjectured for the Casorati determinants associated to the Meixner and Hahn families and the set of numbers –c+i, i = 1, …, k and k,m 0.Artículo Summing Sneddon-Bessel series explicitly(Wiley, 2024-02-09) Durán Guardeño, Antonio José; Pérez, Mario; Varona, Juan Luis; Análisis Matemático; FQM262: Teoría de la AproximaciónArtículo Some examples of orthogonal matrix polynomials satisfying odd order differential equations(Elsevier, 2007-08-29) Durán Guardeño, Antonio José; Iglesia, Manuel D. de la; Análisis Matemático; FQM262: Teoría de la AproximaciónIt is well known that if a finite order linear differential operator with polynomial coefficients has as eigenfunctions a sequence of orthogonal polynomials with respect to a positive measure (with support in the real line), then its order has to be even. This property no longer holds in the case of orthogonal matrix polynomials. The aim of this paper is to present examples of weight matrices such that the corresponding sequences of matrix orthogonal polynomials are eigenfunctions of certain linear differential operators of odd order. The weight matrices are of the form W(t) = tae-teAttBtB*eA*t, where A and B are certain (nilpotent and diagonal, respectively) N X N matrices. These weight matrices are the first examples illustrating this new phenomenon which are not reducible to scalar weights.Artículo Series de productos de senos (y cosenos)(Real Sociedad Matemática Española, 2024) Durán Guardeño, Antonio José; Pérez Riera, Mario; Varona Malumbres, Juan Luis; Análisis Matemático; FQM262: Teoría de la AproximaciónEn este artículo mostramos expresiones explícitas para las series (−1)m+1 (πm)k+2n k j=1 sen(πmxj) con n ∈ N∪{0} y |x1|+|x2|+···+|xk| < 1, o (−1)m ((2m +1)π)k+1+2n k j=1 sen((2m +1)πxj/2), así como algunas otras con cosenos en vez de senos, o con un parámetro adicional.Artículo Ratio asymptotics for orthogonal matrix polynomials(Elsevier, 1999-10) Durán Guardeño, Antonio José; Análisis Matemático; FQM262: Teoría de la AproximaciónRatio asymptotic results give the asymptotic behaviour of the ratio between two consecutive orthogonal polynomials with respect to a positive measure. In this paper, we obtain ratio asymptotic results for orthogonal matrix polynomials and introduce the matrix analogs of the scalar Chebyshev polynomials of the second kind.Artículo Polinomios ortogonales: Una perspectiva de su pasado y una visión de su futuro(Real Academia Sevillana de Ciencias, 1997-04-08) Durán Guardeño, Antonio José; Análisis Matemático; FQM262: Teoría de la AproximaciónArtículo Polinomios con raíces entrelazadas(Universitat Autònoma de Barcelona, 2019) Durán Guardeño, Antonio José; Pérez Riera, Mario; Varona Malumbres, Juan Luis; Análisis Matemático; FQM262: Teoría de la AproximaciónEl objetivo más inmediato de la teoría de aproximación es proporcionar objetos sencillos y fácilmente calculables (polinomios, por ejemplo) que se aproximen a unos objetos dados (funciones de variable real, por ejemplo). Con eso en mente, los polinomios ortogonales son centrales en dicha teoría,y tienen aplicaciones en multitud de campos de la matemática y de la física. Tienen una gran cantidad de propiedades interesantes, y siguen siendo untema de investigación muy activo; de hecho, la investigación sobre polinomios ortogonales ocupa a numerosos matemáticos, y cada año se publican cientos de artículos sobre el tema. Pero no vamos a dedicarnos aquí a ellos, sino que sólo mencionaremos alguna de sus propiedades como justificación para motivar el problema que pretendemos abordar.Artículo Pasiones, piojos, dioses... y matemáticas(Real Sociedad Matemática Española, 2009) Durán Guardeño, Antonio José; Análisis Matemático; FQM262: Teoría de la AproximaciónArtículo Orthogonal polynomials and analytic functions associated to positive definite matrices(Elsevier, 2005-11-10) Berg, Christian; Durán Guardeño, Antonio José; Análisis Matemático; FQM262: Teoría de la AproximaciónFor a positive definite infinite matrix A, we study the relationship between its associated sequence of orthonormal polynomials and the asymptotic behaviour of the smallest eigenvalue of its truncation An of size n X n. For the particular case of A being a Hankel or a Hankel block matrix, our results lead to a characterization of positive measures with finite index of determinacy and of completely indeterminate matrix moment problems, respectively.Artículo Orthogonal matrix polynomials: Zeros and Blumenthal's theorem(Elsevier, 2002-05-25) Durán Guardeño, Antonio José; López Rodríguez, Pedro; Análisis Matemático; FQM262: Teoría de la AproximaciónIn this paper, we establish a quadrature formula and some basic properties of the zeros of a sequence (Pn)nof orthogonal matrix polynomials on the real line with respect to a positive definite matrix of measures. Using these results, we show how to get an orthogonalizing matrix of measures for a sequence (Pn)nsatisfying a matrix three-term recurrence relation. We prove Blumenthal's theorem for orthogonal matrix polynomials describing the support of the orthogonalizing matrix of measures in case the matrix recurrence coefficients associated with these matrix polynomials tend to matrix limits having the same entries on every diagonal.Artículo Orthogonal matrix polynomials, scalar-type Rodrigues' formulas and Pearson equations(Elsevier, 2005-04-20) Durán Guardeño, Antonio José; Grünbaum, F. Alberto; Análisis Matemático; FQM262: Teoría de la AproximaciónSome families of orthogonal matrix polynomials satisfying second-order differential equations with coefficients independent of n have recently been introduced (see [Internat. Math. Res. Notices 10 (2004) 461–484]). An important difference with the scalar classical families of Jacobi, Laguerre and Hermite, is that these matrix families do not satisfy scalar type Rodrigues’ formulas of the type (phipiW)(n)W-1, where is a matrix polynomial of degree not bigger than 2. An example of a modified Rodrigues’ formula, well suited to the matrix case, appears in [Internat. Math. Res. Notices 10 (2004) 482]. In this note, we discuss some of the reasons why a second order differential equation with coefficients independent of n does not imply, in the matrix case, a scalar type Rodrigues’ formula and show that scalar type Rodrigues’ formulas are most likely not going to play in the matrix valued case the important role they played in the scalar valued case. We also mention the roles of a scalar-type Pearson equation as well as that of a noncommutative version of it.Artículo Orthogonal matrix polynomials and quadrature formulas(Elsevier, 2002-02-09) Durán Guardeño, Antonio José; Defez, Emilio; Análisis Matemático; FQM262: Teoría de la AproximaciónWe prove that the nodes of a quadrature formula for a matrix weight with the highest degree of precision must necessarily be the zeros of a certain orthonormal matrix polynomial with respect to the matrix weight and the quadrature coefficients are then the coefficients in the partial fraction decomposition of the ratio between the inverse of this orthonormal matrix polynomial and the associated polynomial of the second kind. We also extend this result for quadrature formulas with degree of precision one unit smaller than the highest possible.Artículo N-extremal matrices of measures for an indeterminate matrix moment problem(Elsevier, 2002-05-25) Durán Guardeño, Antonio José; López Rodríguez, Pedro; Análisis Matemático; FQM262: Teoría de la AproximaciónIn this paper we study the N-extremal matrices of measures associated to a completely indeterminate matrix moment problem, i.e., those matrices of measure W, solutions of a completely indeterminate matrix moment problem for which the linear space of matrix polynomials is dense in the corresponding L2(W).Artículo Measures with finite index of determinacy or a mathematical model for Dr Jekyll and Mr Hyde(AMS, 1997) Berg, Christian; Durán Guardeño, Antonio José; Análisis Matemático; FQM262: Teoría de la AproximaciónIn this note measures with finite index of determinacy (i.e. determinate measures for which there exists a polynomial p such that |p|2 is indeterminate) are characterizated in terms of the operator associated to its Jacobi matrix. Using this characterization, we show that such determinate measures with finite index of determinacy (Jekyll) turn out to be indeterminate (Hyde) when considered as matrices of measures.Artículo Matrix orthogonal polynomials satisfying second-order differential equations: Coping without help from group representation theory(Elsevier, 2007-03-04) Durán Guardeño, Antonio José; Grünbaum, F. Alberto; Análisis Matemático; FQM262: Teoría de la AproximaciónThe method developed in Duran and Grünbaum [Orthogonal matrix polynomials satisfying second order differential equations, Internat. Math. Res. Notices 10 (2004) 461–484] led us to consider polynomials that are orthogonal with respect to weight matrices W(t) of the form e-t2 T(t)T*(t), and ta(1-t)bT(t)T*(t), with T satisfying T' = (2Bt + A)T, T(1) = I and T'(t) = (A/t + B/(1-t)T, T(172), respectively. Here A and B are in general two non-commuting matrices. To proceed further and find situations where these polynomials satisfied second-order differential equations, we needed to impose commutativity assumptions on the pair of matrices A,B. In fact, we only dealt with the case when one of the matrices vanishes. The only exception to this arose as a gift from group representation theory: one automatically gets a situation where A and B do not commute, see Grünbaum et al. [Matrix valued orthogonal polynomials of the Jacobi type: the role of group representation theory, Ann. Inst. Fourier Grenoble 55 (6) (2005) 2051–2068]. This corresponds to the last of the three cases mentioned above. The purpose of this paper is to consider the other two situations and since now we do not get any assistance from representation theory we make a direct attack on certain differential equations in Duran and Grünbaum [Orthogonal matrix polynomials satisfying second order differential equations, Internat. Math. Res. Notices 10 (2004) 461–484]. By solving these equations we get the appropriate weight matrices W(t), where the matrices A,B give rise to a solvable Lie algebra.Artículo Matrix inner product having a matrix symmetric second order differential operator(PROJECT euclid, 1997) Durán Guardeño, Antonio José; Análisis Matemático; FQM262: Teoría de la AproximaciónIn this note we characterize those positive definite matrices of measures whose matricial inner product has a symmetric left-hand side matrix second order differential operator.Artículo Matrix differential equations and scalar polynomials satisfying higher order recursions(Elsevier, 2008-12-24) Durán Guardeño, Antonio José; Grünbaum, F. Alberto; Análisis Matemático; FQM262: Teoría de la AproximaciónWe show that any scalar differential operator with a family of polynomials as its common eigenfunctions leads canonically to a matrix differential operator with the same property. The construction of the corresponding family of matrix valued polynomials has been studied in [A. Durán, A generalization of Favard's theorem for polynomials satisfying a recurrence relation, J. Approx. Theory 74 (1993) 83–109; A. Durán, On orthogonal polynomials with respect to a positive definite matrix of measures, Canad. J. Math. 47 (1995) 88–112; A. Durán, W. van Assche, Orthogonal matrix polynomials and higher order recurrence relations, Linear Algebra Appl. 219 (1995) 261–280] but the existence of a differential operator having them as common eigenfunctions had not been considered. This correspondence goes only one way and most matrix valued situations do not arise in this fashion. We illustrate this general construction with a few examples. In the case of some families of scalar valued polynomials introduced in [F.A. Grünbaum, L. Haine, Bispectral Darboux transformations: An extension of the Krall polynomials, Int. Math. Res. Not. 8 (1997) 359–392] we take a first look at the algebra of all matrix differential operators that share these common eigenfunctions and uncover a number of phenomena that are new to the matrix valued case.