Exploring multivariate Padé approximants for multiple hypergeometric series

We investigate the approximation of some hypergeometric functions of two variables, namely the Appell functions F _i , i = 1,...,4, by multivariate Padé approximants. Section 1 reviews the results that exist for the projection of the F _i onto ϰ=0 or y=0, namely, the Gauss function ₂ F ₁(a, b; c; z), since a great deal is known about Padé approximants for this hypergeometric series. Section 2 summarizes the definitions of both homogeneous and general multivariate Padé approximants. In section 3 we prove that the table of homogeneous multivariate Padé approximants is normal under similar conditions to those that hold in the univariate case. In contrast, in section 4, theorems are given which indicate that, already for the special case F ₁(a, b, b′; c; x; y) with a = b = b′ = 1 and c = 2, there is a high degree of degeneracy in the table of general multivariate Padé approximants. Section 5 presents some concluding remarks, highlighting the difference between the two types of multivariate Padé approximants in this context and discussing directions for future work. This revised version was published online in June 2006 with corrections to the Cover Date.     

N-point Padé approximants to real-valued Stieltjes series with nonzero radii of convergence

By employing special continued fractions to two Stieltjes series with nonzero radii of convergence we extend the inequalities for one-point Padé approximants reported by Baker (1975, Corollory 17.1) to the case of two-point Padé approximants. We prove that some convergents of the continued fractions form a monotone sequences of upper and lower bounds converging uniformly to Stieltjes function x₁(x) on compact subsets of (−R, ∞), where R is a radius of convergence of an expansion of x f₁(x) at x = 0. For an illustration of theoretical results we provide nontrivial numerical examples. As an application to real physical problems second order Padé approximants' bounds on the effective conductivity of a square array of cylinders are evaluated.     

Simultaneous partial Padé approximants

In this paper the concept of partial Padé approximation, introduced by Claude Brezinski, is generalised to the case of simultaneous rational approximation with common denominator. The use of information about known poles and/or zeros, can lead to approximants with a better numerical behaviour than in the case of ordinary simultaneous Padé approximation.     

Convergence of Padé approximants for aq-hypergeometric series (Wynn's Power Series I)

We investigate the convergence of sequences of Padé approximants for power series

Continued fractions,two-point Padé approximants and errors in the Stieltjes case

A Stieltjes function is expanded in mixed T- and S-continued fraction. The relations between approximants of this continued fraction and two-point Padé approximants are established. The method used by Gilewicz and Magnus (J. Comput. Appl. Math. 49 (1993) 79; Integral Transforms Special Functions 1 (1993) 9) has been adapted to obtain the exact relations between the errors of the contiguous two-point Padé approximants in the whole cut complex plane.     

Padé approximants of special functions

Let $ {f_{\gamma }}(x) = \sum\nolimits_{{k = 0}}^{\infty } {{{{T_k (x)}} \left/ {{{{\left( \gamma \right)}_k}}} \right.}} $ , where (??) _k =??(??+1) ? (??+k?1) and T _k (x)=cos (k arccos x) are Padé?CChebyshev polynomials. For such functions and their Padé?CChebyshev approximations $ \pi_{n,m}^{ch}\left( {x;{f_{\gamma }}} \right) $ , we find the asymptotics of decreasing the difference $ {f_{\gamma }}(x) - \pi_{n,m}^{ch}\left( {x;{f_{\gamma }}} \right) $ in the case where 0 ? m ? m(n), m(n) = o (n), as n???? for all x ?? [?1, 1]. Particularly, we determine that, under the same assumption, the Padé?CChebyshev approximations converge to f _?? uniformly on the segment [?1, 1] with the asymptotically best rate.     

Row convergence theorems for generalised inverse vector-valued Padé approximants

The approximants mentioned in the title are related to vector-valued continued fractions and the vector ϵ-algorithm devised by Wynn in 1963. Here we establish a unitary invariance property of these approximants and describe how the classical (1-dimensional) Padé approximants can be obtained as a special case. The main results of the paper consist of De Montessus—De Ballore type convergence theorems for row sequences (having fixed denominator degree) of vector-valued approximants to meromorphic vector functions.     

The convergence of diagonal Padé approximants and the Padé Conjecture

Questions related to the convergence problem of diagonal Padé approximants are discussed. A central place is taken by the Padé Conjecture (also known as the Baker-Gammel-Wills Conjecture). Partial results concerning this conjecture are reviewed and weaker and more special versions of the conjecture are formulated and their plausibility is investigated. Great emphasis is given to the role of spurious poles of the approximants. A conjecture by Nuttall (1970) about the number and distribution of such poles is stated and its importance for the Padé Conjecture is analyzed.     

Padé approximants to certain elliptic-type functions

Among all continua joining non-collinear points a ₁, a ₂, a ₃ ∈ ?, there exists a unique compact Δ ? ? that has minimal logarithmic capacity. For a complex-valued non-vanishing Dini-continuous function h on Δ, we define $${f_h}(z): = \frac{1}{{\pi i}}\int_\Delta {\frac{{h(t)}}{{t - z}}\frac{{dt}}{{{w^ + }(t)}}} $$ , where $w(z): = \sqrt {\prod\nolimits_{k = 0}^3 {(z - {a_k})} } $ and w ⁺ is the one-sided value according to some orientation of 1. In this work, we present strong asymptotics of diagonal Padé approximants to f _h and describe the behavior of the spurious pole and the regions of locally uniform convergence from a generic perspective.     

Universal Padé approximants of Seleznev type

We use the chordal metric in order to approximate all meromorphic functions on \({\mathbb{C} \backslash \{0\}}\) by Padé approximants of formal power series. This is a generic universality of Seleznev type which implies Menchov type almost everywhere approximation with respect to any σ-finite Borel measure on \({\mathbb{C} \backslash \{0\}}\).     

Explicit construction of multivariate Padé approximants

We explicitly construct both homogeneous and nonhomogeneous multivariate Padé approximants to some functions which satisfy some simple functional equations, by using the residue theorem and the functional equation method which has been used successfully by Borwein (1988) to construct one variable Padé approximants to the q-elementary functions.     

Computing high precision Matrix Padé approximants

We describe a new method of computing matrix Padé approximants of series with integer data in an efficient and fraction-free way, by controlling the growth of the size of intermediate coefficients. This algorithm is applied to compute high precision Padé approximants of matrix-valued generating functions of time series. As an illustration we show that we can successfully recover from noisy equidistant sampling data a joint damped signal of four antenna, even in the presence of background signals.     

Inverse homogenization with diagonal Padé approximants

The paper formulates inverse homogenization problem as a problem of recovery of Markov function using diagonal Padé approximants. Inverse homogenization or de-homogenization problem is a problem of deriving information about the micro-geometry of composite material from its effective properties. The approach is based on reconstruction of the spectral measure in the analytic Stieltjes representation of the effective tensor of two-component composite. This representation relates the n-point correlation functions of the microstructure to the moments of the spectral measure, which contains all information about the microgeometry. The problem of identification of the spectral function from effective measurements in an interval of frequency has a unique solution. The problem is formulated as an optimization problem which results in diagonal Padé approximation and exact formulas for the moments of the measure. The reconstructed spectral function can be used to evaluate geometric parameters of the structure and to compute other effective parameters of the same composite; this gives solution to the problem of coupling of different effective properties of a two-component composite material with random microstructure. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Singular rules for the calculation of non-normal multivariate Padé approximants

Section 1 describes the univariate situation in the case of non-normal Padé approximants and Cordellier's extension of the famous five-star identity of Wynn. Section 2 repeats our definition of multivariate Padé approximants and proves a number of theorems that remain valid when going from the univariate to the multivariate case. These theorems and more new results given in Section 3, will finally also copy Cordellier's extension from the univariate to the multivariate case.     

On Padé approximants of Markov-type meromorphic functions

The paper is devoted to the asymptotic properties of diagonal Padé approximants for Markov-type meromorphic functions. The main result is strong asymptotic formulas for the denominators of diagonal Padé approximants for Markov-type meromorphic functions f = \(\hat \sigma \) + r under additional constraints on the measure σ (r is a rational function). On the basis of these formulas, it is proved that, in a sufficiently small neighborhood of a pole of multiplicity m of such a meromorphic function f, all poles of the diagonal Padé approximants f _n are simple and asymptotically located at the vertices of a regular m-gon.     

Root determination by use of Padé approximants

In this paper we describe a new technique for generating iteration formulas — of arbitrary order — for determining a zero (assumed simple) of a functionf, assumed analytic in a region containing the zero. The 1/p Padé Approximant (p0) to the functiong(t)f(z) is formed wherez=w+t, using the Taylor series forf at the pointw, an approxination to the zero off. The value oft for which the 1/p Padé Approximant vanishes provides the basis of iteration formulas of orderp+2.Some known iteration formulas, e.g., Newton-Raphson's, Halley's and Kiss's of order of convergence two, three and four, are directly obtained by settingp=0,1 and 2, respectively.