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.     

Laurent–Padé Approximants to Four Kinds of Chebyshev Polynomial Expansions. Part I. Maehly Type Approximants

Laurent Padé–Chebyshev rational approximants, A _m (z,z ^–1)/B _n (z,z ^–1), whose Laurent series expansions match that of a given function f(z,z ^–1) up to as high a degree in z,z ^–1 as possible, were introduced for first kind Chebyshev polynomials by Clenshaw and Lord [2] and, using Laurent series, by Gragg and Johnson [4]. Further real and complex extensions, based mainly on trigonometric expansions, were discussed by Chisholm and Common [1]. All of these methods require knowledge of Chebyshev coefficients of f up to degree m+n. Earlier, Maehly [5] introduced Padé approximants of the same form, which matched expansions between f(z,z ^–1)B _n (z,z ^–1) and A _m (z,z ^–1). The derivation was relatively simple but required knowledge of Chebyshev coefficients of f up to degree m+2n. In the present paper, Padé–Chebyshev approximants are developed not only to first, but also to second, third and fourth kind Chebyshev polynomial series, based throughout on Laurent series representations of the Maehly type. The procedures for developing the Padé–Chebyshev coefficients are similar to that for a traditional Padé approximant based on power series [8] but with essential modifications. By equating series coefficients and combining equations appropriately, a linear system of equations is successfully developed into two sub-systems, one for determining the denominator coefficients only and one for explicitly defining the numerator coefficients in terms of the denominator coefficients. In all cases, a type (m,n) Padé–Chebyshev approximant, of degree m in the numerator and n in the denominator, is matched to the Chebyshev series up to terms of degree m+n, based on knowledge of the Chebyshev coefficients up to degree m+2n. Numerical tests are carried out on all four Padé–Chebyshev approximants, and results are outstanding, with some formidable improvements being achieved over partial sums of Laurent–Chebyshev series on a variety of functions. In part II of this paper [7] Padé–Chebyshev approximants of Clenshaw–Lord type will be developed for the four kinds of Chebyshev series and compared with those of the Maehly type.     

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.     

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.     

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.     

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 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.     

Hermite–Padé approximation and simultaneous quadrature formulas

We study Hermite–Padé approximation of the so-called Nikishin systems of functions. In particular, the set of multi-indices for which normality is known to take place is considerably enlarged as well as the sequences of multi-indices for which convergence of the corresponding simultaneous rational approximants takes place. These results are applied to the study of the convergence properties of simultaneous quadrature rules of a given function with respect to different weights.     

Simultaneous two-point Padé approximants of Markov functions

By using methods of the potential theory there is studied an asymptotic behavior of simultaneous two-point Padé approximants of two Markov-type functions, one of which is holomorphic in a neighborhood of null, and the other one is holomorphic in the neighborhood of infinity. For special weights, there is obtained the Rodrigue's formula, and there is presented the asymptotics in terms of algebraic functions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 6, pp. 837–841, 1991.     

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.     

Generalized Padé-Type Approximation and Integral Representations

In several complex variables, the multivariate Padé-type approximation theory is based on the polynomial interpolation of the multidimensional Cauchy kernel and leads to complicated computations. In this paper, we replace the multidimensional Cauchy kernel by the Bergman kernel function K (z,x) into an open bounded subset of C ⁿ and, by using interpolating generalized polynomials for K (z,x), we define generalized Padé-type approximants to any f in the space OL ²() of all analytic functions on which are of class L ². The characteristic property of such an approximant is that its Fourier series representation with respect to an orthonormal basis for OL ²() matches the Fourier series expansion of f as far as possible. After studying the error formula and the convergence problem, we show that the generalized Padé-type approximants have integral representations which give rise to the consideration of an integral operator – the so-called generalized Padé-type operator – which maps every f OL ²() to a generalized Padé-type approximant to f. By the continuity of this operator, we obtain some convergence results about series of analytic functions of class L ². Our study concludes with the extension of these ideas into every functional Hilbert space H and also with the definition and properties of the generalized Padé-type approximants to a linear operator of H into itself. As an application we prove a Painlevé-type theorem in C ⁿ and we give two examples making use of generalized Padé-type approximants.     

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.     

Determining radii of meromorphy via orthogonal polynomials on the unit circle

Using a convergence theorem for Fourier–Padé approximants constructed from orthogonal polynomials on the unit circle, we prove an analogue of Hadamard's theorem for determining the radius of m-meromorphy of a function analytic on the unit disk and apply this to the location of poles of the reciprocal of Szeg functions.     

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.     

Rational approximants to symmetric formal Laurent series

Rational approximants, in the Padé sense, to a given formal Laurent series,F(z)= _– c _k z ^k , have been considered by several authors (see [3] for a survey about the different kinds of approximants which can be defined). In this paper, we shall be concerned with symmetric series, that is, when the complex coefficients {c _k } _– ⁺ satisfyc _–k=c _k,k=0, 1,....Making use of Brezinski's approach [1], for Padé-type approximation to a formal power series, rational approximants toF(z) with prescribed poles are obtained, and their algebraic properties considered. These results will allow us to give an alternative approach for the Padé-Chebyshev approximants.     

Computer extension and analytic continuation of problems of porous thrust bearings

Summary In this paper we propose new type of series solution, the semi-analytical, semi-numerical technique for the steady flow of a viscous fluid between two parallel disks in which the fluid is injected through the lower porous disk. We develop a double series expansion of the solution function and sufficiently large number of terms (30 terms-universal coefficients) in the expansion are obtained by delegating routine complex algebra to computer. The expression obtained in the form of power series for the normalized lift is analysed by means of Padé approximants. The change of roles of dependent and independent variables and the use of bilinear Euler transformation increases the region of validity of the corresponding power series. The results obtained from double series expansion method for small as well as moderately large values of cross-flow Reynolds number,R are more accurate and the computing time required in this method is negligible compared with pure numerical methods [1]. Besides this, we find that the method proposed by Phan-Thien and Bush [2] has to be implemented for each value ofR separately, whereas the one proposed by us has advantage of yielding, at a stretch, the results for larger range ofR which agree with exact values and at the same time requires less computer time. It is of interest to note that the pure numerical results in respect of normalized lift coefficient agree closely with the analytic continuation of our findings. In addition various Padé approximants are found to bracket the numerical results.     

An Algorithm for the Computation of Hermite–Padé Approximations to the Exponential Function: Divided Differences and Hermite–Padé Forms

We present the first of two different algorithms for the explicit computation of Hermite–Padé forms (HPF) associated with the exponential function. Some roots of the algebraic equation associated with a given HPF are good approximants to the exponential in some subsets of the complex plane: they are called Hermite–Padé approximants (HPA) to this function. Our algorithm is recursive and based upon the expression of HPF as divided differences of the function texp(xt) at multiple integer nodes. Using this algorithm, we find again the results obtained by Borwein and Driver for quadratic HPF. As an example, we give an interesting family of quadratic HPA to the exponential.