Relative asymptotics and Fourier series of orthogonal polynomials with a discrete Sobolev inner product

21 pages, no figures.-- MSC2000 codes: 42C05, 33C47.MR#: MR1971776 (2004a:42035)Zbl#: Zbl 1014.42019^aLet μ be a finite positive Borel measure supported in [−1,1] and introduce the discrete Sobolev-type inner product $$\langle f,g\rangle = \int^1_{-1} f(x)g(x)d\mu(x)+\sum^K_{k=1} \sum^{N_k}_{i=0} M_{k,i} f^{(i)}(a_k)g^{(i)}(a_k),$$ where the mass points $a_k$ belong to [−1,1], $M_{k,i}\geq 0$, $i = 0,\dots,N_k-1$, and $M_{k,N_k} >0$. In this paper, we study the asymptotics of the Sobolev orthogonal polynomials by comparison with the orthogonal polynomials with respect to the measure μ and we prove that they have the same asymptotic behaviour. We also study the pointwise convergence of the Fourier series associated to this inner product provided that μ is the Jacobi measure. We generalize the work done by F. Marcellán and W. Van Assche where they studied the asymptotics for only one mass point in [−1,1]. The same problem with a finite number of mass points off [−1,1] was solved by G. López, F. Marcellán and W. Van Assche in a more general setting: they consider the constants Mk,i to be complex numbers. As regards the Fourier series, we continue the results achieved by F. Marcellán, B. Osilenker and I.A. Rocha for the Jacobi measure and mass points in R\[-1,1].The work of F. Marcellán was supported by a grant of Dirección General de Investigación (Ministerio de Ciencia y Tecnología) of Spain BFM-2000-0206-C04-01 and INTAS Project, INTAS 00-272.Publicad

Fourier series and orthogonal polynomials in Sobolev spaces

Mención Internacional en el título de doctorIn the last 30 years, the study of orthogonal polynomials in Sobolev spaces has obtained an increasing attention from the research community. The first work on Sobolev orthogonal polynomials was published in 1962 by Althammer, who studied the Legendre-Sobolev polynomials orthogonal with respect to the inner product. The study of this family of orthogonal polynomials is not only interesting for a comparison with the standard theory of orthogonal polynomials, but these polynomials also arise in a natural way in a variety of contexts. In this thesis, we analyze the properties of polynomials orthogonal with respect to a discrete Sobolev inner product. More precisely, we will focus our attention on the study of connection formulas relating Sobolev orthogonal polynomials with the corresponding ordinary ones. Indeed, we deal with some problems on asymptotic behavior of Sobolev orthogonal polynomials as well as we obtain some results on convergence of Fourier-Sobolev series. The present Thesis is organized as follows: In Chapter 1 we introduce the theory of Sobolev orthogonal polynomials and the notation that we will use along this Thesis. We summarize two main differences between the standard orthogonal polynomials and the Sobolev case: recurrence relations and the location of zeros of orthogonal polynomials. Here, we also include a thorough study about the known connection formulas. Finally, for a better understanding of our work, we give the state of the art about asymptotics and Fourier series of orthogonal polynomials, analyzing both the cases of measures with bounded and unbounded support, respectively. In Chapter 2 we study some algebraic and analytic aspects of certain family of Sobolev polynomials orthogonal with respect to a measure with a bounded support on the real line. In Section 2.1 we present an alternative proof for a known result about Outer Relative Asymptotics of Sobolev orthogonal polynomials. In Section 2.2 we also include a new matrix connection relating the matrix associated to the higher order recurrence relation for Sobolev polynomials and the corresponding Jacobi matrix associated to the standard ones. In Section 2.3 we show a result about pointwise convergence of Fourier-Sobolev series in the case of measures with bounded support. In Chapter 3 we summarize some known properties of polynomials orthogonal with respect to a modification of the Laguerre measure, the k-iterated Christoffel one. Later on, we obtain estimates for the norm of such polynomials as well as a generalized Christoffel formula for them. Finally, we present a detailed study about the diagonal Christoffel kernels associated to the Gamma distribution. In particular, we obtain the asymptotic behavior of these kernel polynomials both inside and outside the support of the measure. In Chapter 4 we deal with the Outer and Inner Relative Asymptotics of Sobolevtype orthogonal polynomials when the mass points are located inside the support of the measure, the oscillatory region for such polynomials. Finally, we obtain the asymptotic behavior of the coefficients appearing in the higher order recurrence relation that Sobolev polynomials satisfy. In Chapter 5 we show the divergence of a certain Fourier-Sobolev series. The main tool for this purpose will be a Cohen type inequality. This problem is dealing for the first time for a Sobolev-type inner product with a mass point outside the support of the measure.El desarrollo del estudio de polinomios ortogonales en espacios de Sobolev ha tenido lugar a lo largo de los últimos 30 años. El primer artículo sobre polinomios ortogonales de Sobolev, fue publicado en 1962 por Althammer, quien estudió los polinomios de Sobolev-Legendre ortogonales respecto al producto interno. El estudio de estas nuevas familias de polinomios ortogonales es interesante, no sólo por la comparación entre las propiedades y comportamiento de estos polinomios frente a los polinomios ortogonales, sino por las múltiples aplicaciones que tienen en diferentes contextos. En esta memoria analizaremos el comportamiento y las propiedades de polinomios ortogonales respecto a productos internos de Sobolev discretos. Más concretamente, nuestro interés será el estudio de fórmulas de conexión entre polinomios ortogonales estándar y polinomios ortogonales de Sobolev. De esta forma, podremos abordar algunos problemas de asintótica de polinomios ortogonales de Sobolev, así como obtener resultados de convergencia de series de Fourier asociadas a tales polinomios. Estos contenidos se dividen en los siguientes capítulos: En el capítulo 1 presentamos una introducción a la teoría de polinomios de Sobolev, introduciendo la notación que se utilizaría a lo largo de esta tesis. Se resumirán las principales diferencias entre el caso estándar y el caso Sobolev. En concreto nos centraremos en el estudio de relaciones de recurrencia y localización de ceros de dichos polinomios ortogonales. Se incluirá un estudio bastante completo de los diferentes tipos de fórmulas de conexión existentes. Finalmente, daremos una panorámica de los resultados conocidos en asintótica y desarrollos en series de Fourier de polinomios ortogonales de Sobolev tanto en el caso de soporte acotado como en el no acotado, que permitirá una mejor comprensión de nuestro trabajo. En el capítulo 2 se estudian aspectos analíticos y algebraicos de cierta familia de polinomios de Sobolev ortogonales respecto a una medida de soporte acotado. En la sección 2.1 se presenta una demostración alternativa de un resultado conocido sobre asintótica relativa exterior de ciertos polinomios de Sobolev. Demostraremos una nueva relación matricial entre la matriz asociada a la relación de recurrencia que satisfacen los polinomios de Sobolev y la matriz de Jacobi de los correspondientes polinomios ordinarios en la sección 2.2. En la última sección presentaremos un resultado sobre convergencia puntual de series de Fourier asociadas a ciertos polinomios de Sobolev. En el capítulo 3 resumiremos algunas propiedades conocidas de polinomios ortogonales respecto a una medida de Laguerre modificada, una k-iteración de Christoffel de la medida de Laguerre. A continuación, obtendremos estimaciones para la norma de estos polinomios y proporcionaremos una fórmula generalizada de Christoffel para tal familia de polinomios. Finalmente, presentaremos un estudio completo y detallado de los núcleos de Christoffel diagonales asociados a la distribución Gamma, obteniendo la asintótica de los mismos tanto dentro como fuera del soporte de la medida. En el capítulo 4 se hace un estudio de asintótica relativo de polinomios ortogonales de Sobolev discreto cuando las masas en la parte discreta del producto interno están situadas dentro el soporte de la medida, la región de oscilación de dichos polinomios. El comportamiento de dichos polinomios será estudiado tanto dentro como fuera del soporte de la medida. Finalmente, obtendremos el comportamiento asintótico de los coeficientes que aparecen en la relación de recurrencia que satisfacen los polinomios de Sobolev. En el capítulo 5 abordaremos el problema de convergencia de series de Fourier-Sobolev. Mostraremos la divergencia de la serie de Fourier asociada a cierta familia de polinomios ortogonales de Sobolev y la principal herramienta para ello serán las desigualdades de tipo Cohen. Este problema es tratado por primera vez para un producto de Sobolev discreto con una masa fuera del soporte de la medida.Programa Oficial de Doctorado en Ingeniería MatemáticaPresidente: Guillermo López Lagomasino.- Vocal: M. Alicia Chachafeiro López.- Secretario: José Carlos Soares Petronilh

Asymptotics for Laguerre-Sobolev type orthogonal polynomials modified within their oscillatory regime

In this paper we consider sequences of polynomials orthogonal with respect to the discrete Sobolev inner product (f.g)_s ∫_°^∞ f(x)g(x) x^(α ) e dx+F(c)ΑG(c)^t, α> 1 where f and g are polynomials with real coefficients A∈ R^2.2 and the vectors F(c), G(c) are A=(■(M&0@0&N)), F(c)=(f(c),f'(c) ) G(c)=(g(c),g'(c)) with M,N ∈ R and the mass point c is located inside the oscillatory region for the classical Laguerre polynomials. We focus our attention on the representation of these polynomials in terms of classical Laguerre polynomials and we analyze the behavior of the coefficients of the corresponding five term recurrence relation when the degree of the polynomials is large enough. Also, the outer relative asymptotics of the Laguerre Sobolev type with re spect to the Laguerre polynomials is analyzed

On computational aspects of discrete Sobolev inner products on the unit circle

In this paper, we show how to compute in O(n² )steps the Fourier coefficients associated with the Gelfand-Levitan approach for discrete Sobolev orthogonal polynomials on the unit circle when the support of the discrete component involving derivatives is located outside the closed unit disk. As a consequence, we deduce the outer relative asymptotics of these polynomials in terms of those associated with the original orthogonality measure. Moreover, we show how to recover the discrete part of our Sobolev inner product.The research of K. Castillo was supported by CNPq Program/Young Talent Attraction, Ministério da Ciência, Tecnologia e Inovação of Brazil, Project 370291/2013 1. The research of K. Castillo and F. Marcellán was supported by Dirección General de Investigación, Ministerio de Economía y Competitividad of Spain, Grant MTM2012 36732 C03 01. F. Marcellán also acknowledges the financial support of CAPES Program/Special Visiting Researcher by Ministério da Educação of Brasil, Project 107/201

On Fourier series of a discrete Jacobi-Sobolev inner product

22 pages, no figures.-- MSC2000 code: 42C05.MR#: MR1920116 (2003e:42007)Zbl#: Zbl 1019.42014Let $\mu$ be the Jacobi measure supported on the interval $[-1,1]$ and introduce the discrete Sobolev-type inner product \langle f,g\rangle= \int _{-1} f(x) g(x) d\mu(x)+ \sum _{k=1} \sum N_k}_{i=0} M_{k,i} f (i)}(a_k) g (i)}(a_k), where $a_k$, $1\le k\le K$, are real numbers such that $_k 1$ and $M_{k,i}> 0$ for all $k$, $i$. This paper is a continuation of [{\it F. Marcellán}, {\it B. P. Osilenker} and {\it I. A. Rocha}, "On Fourier series of Jacobi-Sobolev orthogonal polynomials", J. Inequal. Appl. 7, 673-699 (2002; Zbl 1016.42014)] and our main purpose is to study the behaviour of the Fourier series associated with such a Sobolev inner product. For an appropriate function $f$, we prove here that the Fourier-Sobolev series converges to $f$ on $(-1,1)\bigcup _{k=1}\{a_k\}$, and the derivatives of the series converge to f (i)}(a_k) for all $i$ and $k$. Roughly speaking, the term appropriate means here the same as we need for a function $f$ in order to have convergence for its Fourier series associated with the standard inner product given by the measure $\mu$. No additional conditions are needed.The work of F. Marcellán was supported by a grant of Dirección General de Investigación (Ministerio de Ciencia y Tecnología) of Spain BFM 2000 0206 C04 01 and by an INTAS Grant 2000/272.Publicad