On Rational Interpolation to Meromorphic Functions in Several Variables

In this paper, a new approach to construct rational interpolants to functions of several variables is considered. These new families of interpolants, which in fact are particular cases of the so-called Padé-type approximants (that is, rational interpolants with prescribed denominators), extend the classical Padé approximants (for the univariate case) and provide rather general extensions of the well-known Montessus de Ballore theorem for several variables. The accuracy of these approximants and the sharpness of our convergence results are analyzed by means of several examples.     

On the size of multivariate polynomial lemniscates and the convergence of rational approximants

In a previous paper, the author introduced a class of multivariate rational interpolants, which are called optimal Padé-type approximants (OPTA). The main goal of this paper is to extend classical results on convergence both in measure and in capacity of sequences of Padé approximants to the multivariate case using OPTA. To this end, we obtain some estimations of the size of multivariate polynomial lemniscates in terms of the Hausdorff content, which we also think are of some interest.     

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).     

A multivariate QD-like algorithm

The problem of constructing a univariate rational interpolant or Padé approximant for given data can be solved in various equivalent ways: one can compute the explicit solution of the system of interpolation or approximation conditions, or one can start a recursive algorithm, or one can obtain the rational function as the convergent of an interpolating or corresponding continued fraction.In case of multivariate functions general order systems of interpolation conditions for a multivariate rational interpolant and general order systems of approximation conditions for a multivariate Padé approximant were respectively solved in [6] and [9]. Equivalent recursive computation schemes were given in [3] for the rational interpolation case and in [5] for the Padé approximation case. At that moment we stated that the next step was to write the general order rational interpolants and Padé approximants as the convergent of a multivariate continued fraction so that the univariate equivalence of the three main defining techniques was also established for the multivariate case: algebraic relations, recurrence relations, continued fractions. In this paper a multivariate qd-like algorithm is developed that serves this purpose.     

On the structure of a table of multivariate rational interpolants

In the table of multivariate rational interpolants the entries are arranged such that the row index indicates the number of numerator coefficients and the column index the number of denominator coefficients. If the homogeneous system of linear equations defining the denominator coefficients has maximal rank, then the rational interpolant can be represented as a quotient of determinants. If this system has a rank deficiency, then we identify the rational interpolant with another element from the table using less interpolation conditions for its computation and we describe the effect this dependence of interpolation conditions has on the structure of the table of multivariate rational interpolants. In the univariate case the table of solutions to the rational interpolation problem is composed of triangles of so-called minimal solutions, having minimal degree in numerator and denominator and using a minimal number of interpolation conditions to determine the solution.Communicated by Dietrich Braess.     

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 Padé-based algorithm for overcoming the Gibbs phenomenon

Truncated Fourier series and trigonometric interpolants converge slowly for functions with jumps in value or derivatives. The standard Fourier–Padé approximation, which is known to improve on the convergence of partial summation in the case of periodic, globally analytic functions, is here extended to functions with jumps. The resulting methods (given either expansion coefficients or function values) exhibit exponential convergence globally for piecewise analytic functions when the jump location(s) are known. Implementation requires just the solution of a linear system, as in standard Padé approximation. The new methods compare favorably in experiments with existing techniques.     

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.     

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.     

Best choice of the pole for the Padé-type approximant of a Stieltjes function

The aim of this paper is to use Eiermann's theorem to define precisely good poles for the Padé-type approximant of a certain class of functions. Stieltjes functions whose measure has a compact support or functions with a finite number of real singularities are the main examples of this study. The case of an approximant with one multiple pole is completely studied. The case of two poles is considered. Some numerical experiments have been done, showing that the results, obtained by majorization, seem optimal.     

Une construction »homogène» d'approximants de padé à deux variables complexes

Resumé En nous plaçant sur chacune des droites de passant par l'origine, nous utilisons l'interpolation rationnelle d'une seule variable complexe, holomorphe en 0, pour engendrer un certain type d'approximants de Padé dans (x, y).Par cette même démarche, nous essayons de transposer les formules intégrales d'erreur connues pour une seule variable au cadre de deux variables.

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A homogeneous process for padé approximants in two complex variables

Summary By considering the restriction of a function of two complex variables at each straight line passing through the origin and its Padé approximants, we get rational approximants to this function. Integral forms of the error are also used to get convergence results.

     

Extrapolation methods for vector sequences

Summary An analogue of Aitken's ² method, suitable for vector sequences, is proposed. Aspects of the numerical performance of the vector -algorithm, based on using the Moore-Penrose inverse, are investigated. The fact that the denominator polynomial associated with a vector Padé approximant is the square of its equivalent in the scalar case is shown to be a source of approximation error. In cases where the convergence of the vector sequence is dominated by real eigenvalues, a hybrid form of the vector Padé approximant, having a denominator polynomial of minimal degree, is proposed and its effectiveness is demonstrated on several standard examples.     

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.     

The Calculation of Some Rational Approximants in Two Variables

Chisholm has shown how rational approximants may be definedfor functions of two variables, which reduce to diagonal Pad?Approximants when one variable is zero. We extend the idea toallow for different maximum powers in numerator and denominator,and state the relevant theorems. We discuss numerical difficultiesand possible singularities occurring in the construction ofthe approximants; approximants to the Beta function are calculatedand discussed in detail.     

A de Montessus type convergence study for a vector-valued rational interpolation procedure

In a recent paper of the author [8], three new interpolation procedures for vector-valued functions F(z), where F: ℂ → ℂ^N, were proposed, and some of their algebraic properties were studied. In the present work, we concentrate on one of these procedures, denoted IMMPE, and study its convergence properties when it is applied to meromorphic functions. We prove de Montessus and Koenig type theorems in the presence of simple poles when the points of interpolation are chosen appropriately. We also provide simple closed-form expressions for the error in case the function F(z) in question is itself a vector-valued rational function whose denominator polynomial has degree greater than that of the interpolant.     

A method for directly finding the denominator values of rational interpolants

In this paper, we give an algorithm for directly finding the denominator values of rational interpolants at the nodes, and present an expression for the corresponding rational interpolant when the latter exists. With these denominator values, our method also provides information concerning the existence of the interpolant and the presence of unattainable points and poles.     

Matrix Padé Approximation of rational functions

In many applications it is of major interest to decide whether a given formal power series with matrix-valued coefficients of arbitrary dimensions results from a matrix-valued rational function. As the main result of this paper we provide an answer to this question in terms of Matrix Padé Approximants of the given power series. Furthermore, given a matrix rational function, the smallest degrees of the matrix polynomials which represent it are not necessarily unique. Therefore we study a certain minimality-type, that is, minimum degrees. We aim to obtain all the minimum degrees for the polynomials which represent the function as equivalents. In addition, given that the rational representation of the function for the same pair of degrees need not be unique, we have obtained conditions to study the uniqueness of said representation. All the results obtained are presented graphically in tables setting out the above information. They lead to a number of properties concerning special structures, staired blocks, in the Padé Table.     

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.     

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.     

Direct and Inverse Results on Row Sequences of Hermite–Padé Approximants

We give necessary and sufficient conditions for the convergence with geometric rate of the common denominators of simultaneous rational interpolants with a bounded number of poles. The conditions are expressed in terms of intrinsic properties of the system of functions used to build the approximants. Exact rates of convergence for these denominators and the simultaneous rational approximants are provided.