Systems of Markov type functions: normality and convergence of Hermite-Padé approximants

This thesis deals with Hermite-Padé approximation of analytic and merophorphic functions. As such it is embeded in the theory of vector rational approximation of analytic functions which in turn is intimately connectd with the theory of multiple orthogonal polynomials. All the basic concepts and results used in this thesis involving complex analysis and measure theory may found in classical textbooks...........Programa Oficial de Doctorado en Ingeniería MatemáticaPresidente: Francisco José Marcellán Español; Vocal: Alexander Ivanovich Aptekarev; Secretario: Andrei Martínez Finkelshtei

Convergence and Asymptotic of Multi-Level Hermite-Padé Polynomials

Mención Internacional en el título de doctorPrograma de Doctorado en Ingeniería Matemática por la Universidad Carlos III de MadridPresidente: Francisco José Marcellán Español.- Secretario: Bernardo de la Calle Ysern.- Vocal: Arnoldus Bernardus Jacobus Kuijla

The Devil's Invention: Asymptotic, Superasymptotic and Hyperasymptotic Series

Singular perturbation methods, such as the method of multiple scales and the method of matched asymptotic expansions, give series in a small parameter ε which are asymptotic but (usually) divergent. In this survey, we use a plethora of examples to illustrate the cause of the divergence, and explain how this knowledge can be exploited to generate a 'hyperasymptotic' approximation. This adds a second asymptotic expansion, with different scaling assumptions about the size of various terms in the problem, to achieve a minimum error much smaller than the best possible with the original asymptotic series. (This rescale-and-add process can be repeated further.) Weakly nonlocal solitary waves are used as an illustration.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/41670/1/10440_2004_Article_193995.pd

Multipoint Padé Approximants to Complex Cauchy Transforms with Polar Singularities

International audienceWe study diagonal multipoint Padé approximants to functions of the form F(z) = \int\frac{d\mes(t)}{z-t}+R(z), where $R$ is a rational function and \mes is a complex measure with compact regular support included in $\R$, whose argument has bounded variation on the support. Assuming that interpolation sets are such that their normalized counting measures converge sufficiently fast in the weak-star sense to some conjugate-symmetric distribution $\sigma$, we show that the counting measures of poles of the approximants converge to $\widehat\sigma$, the balayage of $\sigma$ onto the support of \mes, in the weak$^*$ sense, that the approximants themselves converge in capacity to $F$ outside the support of \mes, and that the poles of $R$ attract at least as many poles of the approximants as their multiplicity and not much more

Uniformization and Constructive Analytic Continuation of Taylor Series

We analyze the general mathematical problem of global reconstruction of a function with least possible errors, based on partial information such as n terms of a Taylor series at a point, possibly also with coefficients of finite precision. We refer to this as the "inverse approximation theory problem, because we seek to reconstruct a function from a given approximation, rather than constructing an approximation for a given function. Within the class of functions analytic on a common Riemann surface Omega, and a common Maclaurin series, we prove an optimality result on their reconstruction at other points on Omega, and provide a method to attain it. The procedure uses the uniformization theorem, and the optimal reconstruction errors depend only on the distance to the origin. We provide explicit uniformization maps for some Riemann surfaces Omega of interest in applications. One such map is the covering of the Borel plane of the tritronquee solutions to the Painleve equations PI-PV. As an application we show that this uniformization map leads to dramatic improvement in the extrapolation of the PI tritronquee solution throughout its domain of analyticity and also into the pole sector. Given further information about the function, such as is available for the ubiquitous class of resurgent functions, significantly better approximations are possible and we construct them. In particular, any one of their singularities can be eliminated by specific linear operators, and the local structure at the chosen singularity can be obtained in fine detail. More generally, for functions of reasonable complexity, based on the nth order truncates alone we propose new efficient tools which are convergent as n to infty, which provide near-optimal approximations of functions globally, as well as in their most interesting regions, near singularities or natural boundaries.Comment: 39 pages, 9 figures; v2 some clarifications adde

Ratio and relative asymptotics of polynomials orthogonal with respect to varying Denisov-type measures

34 pages, no figures.-- MSC1991 codes: 42C05, 41A28.-- Dedicated to Barry Simon on the occasion of his sixtieth birthday.MR#: MR2220040 (2006m:42002)Zbl#: Zbl 1100.42014Let μ be a finite positive Borel measure with compact support consisting of an interval [c,d] ⊂ R plus a set of isolated points in R\[c,d], such that μ′>0 almost everywhere on [c,d]. Let $\{w_{2n}\}$, $n\in\Bbb Z_+$, be a sequence of polynomials, $\deg w_{2n}\leq2n$, with real coefficients whose zeros lie outside the smallest interval containing the support of μ. We prove ratio and relative asymptotics of sequences of orthogonal polynomials with respect to varying measures of the form $\frac{d\mu_n}{w_{2n}}$. In particular, we obtain an analogue for varying measures of Denisov's extension of Rakhmanov's theorem on ratio asymptotics. These results on varying measures are applied to obtain ratio asymptotics for orthogonal polynomials with respect to fixed measures on the unit circle and for multi-orthogonal polynomials in which the measures involved are of the type described above.Research of D.B. Rolanía was partially supported by Dirección General de Investigación, Ministerio de Ciencia y Tecnología, under Grant BFM 2003-06335-C03-02. Research of de la Calle Ysern was supported by Dirección General de Investigación, Ministerio de Ciencia y Tecnología, under Grants BFM 2002-04315-C02-01 and BFM 2003-06335-C03-02. Research of G.L. Lagomasino was supported by Grants INTAS 03-516637, NATO PST.CLG.979738, and by Dirección General de Investigación, Ministerio de Ciencia y Tecnología, under Grant BFM 2003-06335-C03-02.Publicad

An Lp Analog to AAK Theory for p⩾2

AbstractWe develop an Lp analog to AAK theory on the unit circle that interpolates continuously between the case p=∞, which classically solves for best uniform meromorphic approximation, and the case p=2, which is equivalent to H2-best rational approximation. We apply the results to the uniqueness problem in rational approximation and to the asymptotic behaviour of poles of best meromorphic approximants to functions with two branch points. As pointed out by a referee, part of the theory extends to every p∈[1, ∞] when the definition of the Hankel operator is suitably generalized; this we discuss in connection with the recent manuscript by V. A. Prokhorov, submitted for publication

Tosio Kato’s work on non-relativistic quantum mechanics: part 1

We review the work of Tosio Kato on the mathematics of non-relativistic quantum mechanics and some of the research that was motivated by this. Topics in this first part include analytic and asymptotic eigenvalue perturbation theory, Temple–Kato inequality, self-adjointness results, and quadratic forms including monotone convergence theorems

Spectral problems and orthogonal polynomials on the unit circle

The main purpose of the work presented here is to study transformations of sequences of orthogonal polynomials associated with a hermitian linear functional L, using spectral transformations of the corresponding C-function F. We show that a rational spectral transformation of F is given by a finite composition of four canonical spectral transformations. In addition to the canonical spectral transformations, we deal with two new examples of linear spectral transformations. First, we analyze a spectral transformation of L such that the corresponding moment matrix is the result of the addition of a constant on the main diagonal or on two symmetric sub-diagonals of the initial moment matrix. Next, we introduce a spectral transformation of L by the addition of the first derivative of a complex Dirac linear functional when its support is a point on the unit circle or two points symmetric with respect to the unit circle. In this case, outer relative asymptotics for the new sequences of orthogonal polynomials in terms of the original ones are obtained. Necessary and su cient conditions for the quasi-definiteness of the new linear functionals are given. The relation between the corresponding sequence of orthogonal polynomials in terms of the original one is presented. We also consider polynomials which satisfy the same recurrence relation as the polynomials orthogonal with respect to the linear functional L , with the restriction that the Verblunsky coe cients are in modulus greater than one. With positive or alternating positive-negative values for Verblunsky coe cients, zeros, quadrature rules, integral representation, and associated moment problem are analyzed. We also investigate the location, monotonicity, and asymptotics of the zeros of polynomials orthogonal with respect to a discrete Sobolev inner product for measures supported on the real line and on the unit circle. -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------El objetivo principal de este trabajo es el estudio de las sucesiones de polinomios ortogonales con respecto a transformaciones de un funcional lineal hermitiano L , usando para ello las transformaciones de la correspondiente C-función F . Un primer resultado es que las transformaciones espectrales racionales de F están dadas por una composición finita de cuatro transformaciones espectrales canónicas. Además de estas transformaciones canónicas se estudian dos ejemplos de transformaciones espectrales lineales que son novedosos en la literatura. El primero de estos ejemplos está dado por una modificación del funcional lineal L, de modo que la correspondiente matriz de momentos es el resultado de la adición de una constante en la diagonal principal o en dos subdiagonales simétricas de la matriz de momentos original. El segundo ejemplo es una transformación de L mediante la adición de la primera derivada de una delta de Dirac compleja cuando su soporte es un punto sobre la circunferencia unidad o dos puntos simétricos respecto a la circunferencia unidad. En este caso se obtiene la asintótica relativa exterior de la nueva sucesión de polinomios ortogonales en términos de la original. Se dan condiciones necesarias y suficientes para que los funcionales derivados de las perturbaciones estudiadas sean cuasi-definidos, y se obtiene la relación entre las correspondientes sucesiones de polinomios ortogonales. Se consideran además polinomios que satisfacen las mismas ecuaciones de recurrencia que los polinomios ortogonales con respecto al funcional lineal L, agregando la restricción de que sus coeficientes de Verblunsky son en valor absoluto mayores que 1. Cuando estos coeficientes son positivos o alternan signo, se estudian los ceros, las fórmulas de cuadratura, la representación integral y el problema de momentos asociado. Asimismo, se estudia la localización, monotonicidad y comportamiento asintótico de los ceros asociados a polinomios discretos ortogonales de Sobolev para medidas soportadas tanto en la recta real como en la circunferencia unidad.This work was supported by FPU Research Fellowships, Re. AP2008-00471 and Dirección General de Investigación, Ministerio de Ciencia e Inovación of Spain, grant MTM2009-12740-C03-01