IOPA seminar10/11/2020
Speaker: Ulises Fidalgo.
Title: Nikishin systems and Hermite-Padé approximation.
Abstract: Attending a request from Professor Ranga this Seminar will deal on algebraic and analytic properties of Hermite-Padé approximants corresponding to Nikishin systems. Example: perfectness and asymptotic behavior in different sense, such as in capacity and in n-th roots.
27/10/2020
Speaker: Edmundo Huertas Cejudo
Title: A generalization of Jacobi continued fractions for non-standard Sobolev-type orthogonal polynomials.
Abstract: In the celebrated Chihara’s book on orthogonal polynomials, one can finds (Chapter III) a beautiful relation between standard orthogonal polynomials and continued fractions. This relation flows through the coefficients of the 3TRR satisfied by (all) the standard orthogonal polynomials sequences. Using certain three term recurrence formula with rational coefficients, satisfied by a good number of non-standard Sobolev-type orthogonal sequences, we find a simple extension of the aforesaid relation between orthogonal polynomials and continued fractions. We present a particular example using Al-Salam–Carlitz I-Sobolev-type orthogonal polynomials of higher order. This is a joint work with Anier Soria-Lorente, Alberto Lastra and Carlos Hermoso.
13/10/2020
Speaker: Herbert Dueñas Ruiz
Title: Some ideas about orthogonal polynomials of several variables.
Abstract: I present some ideas that we have been studying about Sobolev orthogonal polynomials in several variables. Some of them are results obtained in [1] and [2], but also, I want to show some results obtained by Yuan Xu in [3] and [4], about orthogonal polynomials in and on quadratic surfaces
References:
[1] Dueñas H and Salazar-Morales O., Laguerre-Gegenbauer-Sobolev orthogonal polynomials in two variables on product domains. Revis. Col. Mat. Accepted.
[2] Dueñas H, Piñar M and Salazar-Morales O., Sobolev orthogonal polynomials of several variables on product domains. (Submitted)
[3] Xu Y., Orthogonal polynomials and Fourier orthogonal series on a cone, J. Fourier Anal. Appl. 26. (2020).
[4] S. Olver and Xu Y., Orthogonal polynomials in and on a quadratic surface of revolution. Comp. Math. Accepted.
15/9/2020
Speaker: Lino Gustavo Garza Gaona
Title: On Differential Equations Associated with Perturbations of Orthogonal Polynomials on the Unit Circle.
Abstract: In this contribution, we propose an algorithm to compute holonomic second-order differential equations satisfied by some families of orthogonal polynomials. Such an algorithm is based on three properties that orthogonal polynomials satisfy: a recurrence relation, a structure formula, and a connection formula. This approach is used to obtain second-order differential equations whose solutions are orthogonal polynomials associated with some spectral transformations of a measure on the unit circle, as well as orthogonal polynomials associated with coherent pairs of measures on the unit circle.
18/8/2020
Speaker: Cleonice Bracciali
Title: Computing quadrature formulae on the real line and on the unit circle.
Abstract: In this talk we consider the theoretical and numerical aspects of the quadrature rules associated with sequences of polynomials generated by a special $R_{II}$ recurrence relation. We look into some methods for generating the nodes (which lie on the real line) and the positive weights of these quadrature rules. With a transformation these quadrature rules on the real line also lead to certain positive quadrature rules of highest algebraic degree of precision on the unit circle. Joint work with J.A. Pereira and A. Sri Ranga.
https://drive.google.com/file/d/1H_nGvy15xdT20Lw6nZNt0B9fLmsjk0f0/view?usp=sharing
1/9/2020
Speaker: Jorge Borrego
Title: On a class of para-orthogonal polynomials associated to a Schrödinger equation with energy dependent potential.
Abstract: We study a class of para-orthogonal polynomials associated to a family of hypergeometric orthogonal polynomials on the unit circle which define explicit solutions of an energy dependent potential Schrödinger equation. The potential contains, as a particular case, the symmetric Rosen–Morse potential. We show analytic properties, asymptotic behavior and relations of orthogonality of the eigenfunctions of the energy dependent Schrödinger equation.
References:
Borrego-Morell, J.A.; Bracciali, C.F.; Sri Ranga, A. On an Energy-Dependent Quantum System with Solutions in Terms of a Class of Hypergeometric Para-Orthogonal Polynomials on the Unit Circle. Mathematics 2020, 8, 1161.
1/9/2020
Speaker: Swaminathan Anbu
Title: Orthogonal Polynomials Associated with perturbed Chain Sequences.
Abstract: Using the minimal parameter sequence of a given chain sequence, we introduce the concept of complementary chain sequences, which we view as perturbations of chain sequences. Using the relation between these complementary chain sequences and the corresponding Verblunsky coefficients, the para-orthogonal polynomials and the associated Szego polynomials are analyzed. Two illustrations, one involving Gaussian hypergeometric functions and the other involving Caratheodory functions are also provided. A connection between these two illustrations by means of complementary chain sequences is also observed.
https://drive.google.com/file/d/1ZCPY03Fqi6xf4TmdpnWCAhI_cFefBg80/view?usp=sharing
4/8/2020
Speaker: Luis Garza
Title: Stability of linear systems and orthogonal polynomials.
Abstract: It is well known that orthogonal polynomials on the real line and orthogonal on the unit circle are related to Hurwitz polynomials and Schur polynomials, respectively. These latter polynomials are referred to as stable polynomials, and characterize the stability of continuous and discrete linear systems. In this talk, we present some new aspects of these relations, as well as some applications in robust stability of linear systems.
https://drive.google.com/file/d/1MlEKIknyV1WJFVcL74QsNHaXj0B4pZRo/view?usp=sharing
21/7/2020
Speaker: Ulises Fidalgo
Title: Stars, Buses (guaguas), and multiple-orthogonal polynomials.
Abstract: We consider a stationary Markov process that models certain queues with a bulk service of fixed number $m$ of admitted customers. We give an integral expression of its transition probability function in terms of certain multi-orthogonal polynomials. We give an algorithm for computing this integral expression, with some examples.
https://drive.google.com/file/d/1JxTWgesESHaYffQjnudNLnT96pQootWB/view?usp=sharing