On the convergence of type I Hermite–Padé approximants

Padé approximation has two natural extensions to vector rational approximation through the so-called type I and type II Hermite–Padé approximants. The convergence properties of type II Hermite–Padé approximants have been studied. For such approximants Markov and Stieltjes type theorems are available. To the present, such results have not been obtained for type I approximants. In this paper, we provide Markov and Stieltjes type theorems on the convergence of type I Hermite–Padé approximants for Nikishin systems of functions.     

Multivariate Padé approximants revisited

Several definitions of multivariate Padé approximants have been introduced during the last decade. We will here consider all types of definitions based on the choice that the coefficients in numerator and denominator of the multivariate Padé approximant are defined by means of a linear system of equations. In this case a determinant representation for the multivariate Padé approximant exists. We will show that a general recursive algorithm can be formulated to compute a multivariate Padé approximant given by any definition of this type. Here intermediate results in the recursive computation scheme will also be multivariate Padé approximants. Up to now such a recursive computation of multivariate Padé approximants only seemed possible in some special cases.     

On two-point Padé-Type and two-Point Padé approximants

Summary Two-point Padé-type approximants are introduced in the case of a non-commutative algebra on a commutative field. Algebraic properties are given and a study of the error of approximation is done. From the relation of the error and some additional properties, two-point Padé approximants are found. Algebraic properties and recurrence relations are proved. The means to compute these approximants in following any way in the table of the approximants are given. The mixed table is introduced, as well as the normality and some results of convergence of two-point Padé-type and Padé approximants.     

Generalized moment representations and Padé-Chebyshev approximations

An approach to the application of Dzyadyk's generalized moment representations in problems of construction and investigation of the Padé-Chebyshev approximants is developed. With its help, certain properties of the Padé-Chebyshev approximants of a class of functions that is a natural analog of the class of Markov functions are studied. In particular, it is proved that the poles of the Padé-Chebyshev approximants of these functions lie outside their domain of analyticity.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 6, pp. 762–766, June, 1990.     

Convergence Rates of Padé and Padé-Type Approximants

A comparison is made between Padé and Padé-type approximants. LetQ_nbe thenth orthonormal polynomial with respect to a positive measureμwith compact support inC. We show that for functions of the form[formula]wherewis an analytic function on the support ofμ, Padé-type approximants with denominatorQ_ngive a successful and, in general, better approximation procedure than Padé approximation.     

Defects and the Convergence of Padé Approximants

A selective survey is given of convergence results for sequences of Padé approximants. Various approaches for dealing with the convergence problems due to `defects" are discussed. Attention is drawn to the close relationship between analyticity properties of a function and the `smoothness" of its Taylor series coefficients. A new theorem on the convergence of horizontal sequences of Padé approximants to functions in the Baker–Gammel–Wills conjecture function class is presented.     

Multivariate Two-Point Padé-Type and Two-Point Padé Approximants

In his paper the notions of two-point Padé-type and two-point Padé approximants are generalized for multivariate functions, with a generating denominator polynomial of general form. The multivariate two-point Padé approximant can be expressed as a ratio of two determinants and computed recursively using the E-algorithm. A comparison is made with previous definitions by other authors using particular generating denominator polynomials. The last section contains some convergence results.     

A direct approach to convergence of multivariate,nonhomogeneous, Padé approximants

We present a direct approach for proving convergence in measure/product capacity of multivariate, nonhomogeneous, Padé approximants. Previous approaches have involved projection onto Padé-type approximation in one variable, and only yielded convergence in (Lebesgue) measure.     

Anti-Gaussian Padé Approximants

This paper gives a synthesis of Padé approximants and anti-Gaussian quadratures. New rational approximants for Stieltjes series have been constructed. In addition, a three term recurrence relation is given for the numerator and denominator, which is useful when the given functional is not defin ite positive.We give the different algebraic properties of these new polynomials, which are similar to those obtained with the Gaussian quadrature formula. We find an easy definition and several relations with Padé approximants. Finally, some numerical results are given in the last section.     

Two-point Padé approximants for formal Stieltjes series

We prove that certain two-point Padé approximants occupying the diagonal of the Padé table form monotone sequences of lower and upper bounds uniformly converging to a Stieltjes function. The results can be applied to the theory of inhomogeneous media for the calculation of the bounds on the effective transport coefficients of heterogeneous materials.     

Multivariate Frobenius–Padé approximants: Properties and algorithms

The aim of this paper is to construct rational approximants for multivariate functions given by their expansion in an orthogonal polynomial system. This will be done by generalizing the concept of multivariate Padé approximation. After defining the multivariate Frobenius–Padé approximants, we will be interested in the two following problems: the first one is to develop recursive algorithms for the computation of the value of a sequence of approximants at a given point. The second one is to compute the coefficients of the numerator and denominator of the approximants by solving a linear system. For some particular cases we will obtain a displacement rank structure for the matrix of the system we have to solve. The case of a Tchebyshev expansion is considered in more detail.     

Some convergence results for the generalized Padé-type approximants

The aim of this paper is to give some convergence results for some sequences of generalized Padé-type approximants. We will consider two types of interpolatory functionals: one corresponding to Langrange and Hermite interpolation and the other corresponding to orthogonal expansions. For these two cases we will give sufficient conditions on the generating functionG(x, t) and on the linear functionalc in order to obtain the convergence of the corresponding sequence of generalized Padé-type approximants. Some examples are given.     

A Rational Approximant for the Digamma Function

Power series representations for special functions are computationally satisfactory only in the vicinity of the expansion point. Thus, it is an obvious idea to use Padé approximants or other rational functions constructed from sequence transformations instead. However, neither Padé approximants nor sequence transformation utilize the information which is avaliable in the case of a special function – all power series coefficients as well as the truncation errors are explicitly known – in an optimal way. Thus, alternative rational approximants, which can profit from additional information of that kind, would be desirable. It is shown that in this way a rational approximant for the digamma function can be constructed which possesses a transformation error given by an explicitly known series expansion.     

Nuttall-Pommerenke theorems for homogeneous Padé approximants

We investigate Nuttall-Pommerenke theorems for several variable homogeneous Padé approximants using ideas of Goncar, Karlsson and Wallin.     

Analytic continuation of Taylor series and the boundary value problems of some nonlinear ordinary differential equations

We compare and discuss the respective efficiency of three methods (with two variants for each of them), based respectively on Taylor (Maclaurin) series, Padé approximants and conformal mappings, for solving quasi-analytically a two-point boundary value problem of a nonlinear ordinary differential equation (ODE). Six configurations of ODE and boundary conditions are successively considered according to the increasing difficulties that they present. After having indicated that the Taylor series method almost always requires the recourse to analytical continuation procedures to be efficient, we use the complementarity of the two remaining methods (Padé and conformal mapping) to illustrate their respective advantages and limitations. We emphasize the importance of the existence of solutions with movable singularities for the efficiency of the methods, particularly for the so-called Padé-Hankel method. (We show that this latter method is equivalent to pushing a movable pole to infinity.) For each configuration, we determine the singularity distribution (in the complex plane of the independent variable) of the solution sought and show how this distribution controls the efficiency of the two methods. In general the method based on Padé approximants is easy to use and robust but may be awkward in some circumstances whereas the conformal mapping method is a very fine method which should be used when high accuracy is required.     

Formal Orthogonal Polynomials and Newton–Padé Approximants

We proved in [8] that the denominators of Newton–Padé approximants for a formal Newton series are formal orthogonal with respect to linear functionals. The same functional is used along an antidiagonal of the Newton–Padé denominator table. The two linear functionals, corresponding to two adjacent antidiagonals, are linked with a very simple relation. Recurrence relations between denominators are given along an antidiagonal or two adjacent antidiagonals in the normal and non-normal case. The same recurrence relations are also satisfied by the Newton–Padé numerators, which implies another formal orthogonality.     

On interpolatory multivariate Padé-type approximants

It is proved that higher order interpolatory Padé-type approximants in two variables do not exist.     

Approximants de pade-hermite. 1ère partie: theorie

Summary We present in this first paper a generalization of Padé approximants which gives us as particular cases Shafer's and Baker'sD-log approximants.First we define these approximants following an old idea of Hermite, then we prove some fundamental properties for their constructions.

     

A type of matrix Padé approximant inspired by scalar component models

In this paper we define a type of matrix Padé approximant inspired by the identification stage of multivariate time series models considering scalar component models. Of course, the formalization of certain properties in the matrix Padé approximation framework can be applied to time series models and in other fields. Specifically, we want to study matrix Padé approximants as follows: to find rational representations (or rational approximations) of a matrix formal power series, with both matrix polynomials, numerator and denominator, satisfying three conditions: (a) minimum row degrees for the numerator and denominator, (b) an invertible denominator at the origin, and (c) canonical representation (without free parameters).