Rational approximation to harmonic functions

Padé-type approximation is the rational function analogue of Taylor’s polynomial approximation to a power series. A general method for obtaining Padé-type approximants to Fourier series expansions of harmonic functions is defined. This method is based on the Newton-Cotes and Gauss quadrature formulas. Several concrete examples are given and the convergence behavior of a sequence of such approximants is studied. This revised version was published online in June 2006 with corrections to the Cover Date.     

Rational interpolation and best rational approximation

A relation between the degree of convergence (in capacity) of Padé approximants and the degree of best rational approximation is derived for functions in Gon?ar's class R₀. The results contain previous convergence results for Padé approximants proved by Nuttall, Pommerenke, and Gammel and Nuttall.     

Convergent Interpolation to Cauchy Integrals over Analytic Arcs

We consider multipoint Padé approximation to Cauchy transforms of complex measures. We show that if the support of a measure is an analytic Jordan arc and if the measure itself is absolutely continuous with respect to the equilibrium distribution of that arc with Dini-smooth nonvanishing density, then the diagonal multipoint Padé approximants associated with appropriate interpolation schemes converge locally uniformly to the approximated Cauchy transform in the complement of the arc. This asymptotic behavior of Padé approximants is deduced from the analysis of underlying non-Hermitian orthogonal polynomials, for which we use classical properties of Hankel and Toeplitz operators on smooth curves. A construction of the appropriate interpolation schemes is explicit granted the parametrization of the arc.     

On the phenomenon of spurious interpolation of elliptic functions by diagonal Padé approximants

We study diagonal Padé approximants for elliptic functions. The presence of spurious poles in the approximants not corresponding to the singularities of the original function prevents uniform convergence of the approximants in the Stahl domain. This phenomenon turns out to be closely related to the existence in the Stahl domain of points of spurious interpolation at which the Padé approximants interpolate the other branch of the elliptic function. We also investigate the behavior of diagonal Padé approximants in a neighborhood of points of spurious interpolation.     

Kronecker type theorems,normality and continuity of the multivariate Padé operator

Summary. For univariate functions the Kronecker theorem, stating the equivalence between the existence of an infinite block in the table of Padé approximants and the approximated function being rational, is well-known. In [Lubi88] Lubinsky proved that if is not rational, then its Padé table is normal almost everywhere: for an at most countable set of points the Taylor series expansion of is such that it generates a non-normal Padé table. This implies that the Padé operator is an almost always continuous operator because it is continuous when computing a normal Padé approximant [Wuyt81]. In this paper we generalize the above results to the case of multivariate Padé approximation. We distinguish between two different approaches for the definition of multivariate Padé approximants: the general order one introduced in [Levi76, CuVe84] and the so-called homogeneous one discussed in [Cuyt84]. Received December 19, 1994     

On nonstandard Padé approximants suitable for effective properties of two-phase composite materials

This article investigates the existence of the nonstandard Padé approximants introduced by Cherkaev and Zhang [D.-L. Zhang and E. Cherkaev, Reconstruction of spectral function from effective permittivity of a composite material using rational function approximations, J. Comput. Phys. 228 (2009), pp. 5390–5409] for approximating the spectral function of composites from effective properties at different frequencies. The spectral functions contain information on microstructure of composites. Since this reconstruction problem is ill-posed Cherkaev [Inverse homogenization for evaluation of effective properties of a mixture, Inverse Probl. 17 (2001), pp. 1203–1218], the well-performed Padé approach is noteworthy and deserves further investigations. In this article, we validate the assumption that the effective dielectric component of interest of all two-phase composites can be approximated by Padé approximants whose denominator has nonzero power one term. We refer to this as the nonstandard Padé approximant, in contrast to the standard approximants whose denominators have nonzero constant terms. For composites whose spectral function assumes infinitely many different values such as the checkerboard microstructure, the proof is carried by using classical results for Markov–Stieltjes functions (also referred to as Stieltjes functions) Golden and Papanicolaou [Bounds on effective parameters of heterogeneous media by analytic continuation, Commun. Math. Phys. 90 (1983), pp. 473–491] and Cherkaev and Ou [De-homogenization: Reconstruction of moments of the spectral measure of the composite, Inverse Probl. 24 (2008), p. 065008]. However, it is well-known that spectral functions for microstructure such as rank-n laminates assume only finitely many different values, i.e. the measure in the Markov–Stieltjes function is supported at only finitely many points. For this case, we cannot find any existence results for nonstandard Padé approximants in the literature. The proof for this case is the focus of this article. It is done by utilizing a special product decomposition of the coefficient matrix of the Padé system. The results in this article can be considered as an extension of the Padé theory for Markov–Stieltjes functions whose spectral function take infinitely many different values to those taking only finitely many values. In the literature, the latter is usually excluded from the definition of Markov–Stieltjes functions because they correspond to rational functions, hence convergence of their Padé approximants is trivial. However, from an inverse problem point of view, we need to assure both the existence and convergence of the nonstandard Padé approximants, for all microstructures. The results in this article provide a mathematical foundation for applying the Padé approach for reconstructing the spectral functions of composites whose microstructure is not a priori known.     

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.     

Universal Padé approximants with respect to the chordal metric

The use of the chordal distance does not change the notion of Universal Taylor series. However, it changes the notion of Universal Padé approximants. Using Padé approximants of meromorphic or holomorphic functions we can approximate all rational functions on compact sets of arbitrary connectivity.     

Padé-Faber approximation of Markov functions on real-symmetric compact sets

Study of Padé-Faber approximation (generalization of Padéapproximation and Padé-Chebyshev approximation) of Markov functions is important not only from the point of view of mathematical analysis, but also of computational mathematics. The theorem on the existence of subdiagonal approximants is constructively proved. Various estimates of the approximation error are given. Theoretical assertions are illustrated by simulation results.     

On the convergence of generalized Padé approximants

Letf be an analytic function with all its singularities in a compact set \(E_f \subset \bar C\) of (logarithmic) capacity zero. The function may have branch points. The convergence of generalized (multipoint) Padé approximants to this type of function is investigated. For appropriately selected schemes of interpolation points, it is shown that close-to-diagonal sequences of generalized Padé approximants converge in capacity tof in a certain domain that can be characterized by the property of the minimal condenser capacity. Using a pole elimination procedure, another set of rational approximants tof is derived from the considered generalized Padé approximants. These new approximants converge uniformly on a given continuum \(V \subset \bar C\backslash E_f\) with a rate of convergence that has been conjectured to be best possible. The continuumV is assumed not to divide the complex plane.     

On the Froissart phenomenon in multivariate homogeneous Padé approximation

In univariate Padé approximation we learn from the Froissart phenomenon that Padé approximants to perturbed Taylor series exhibit almost cancelling pole–zero combinations that are unwanted. The location of these pole–zero doublets was recently characterized for rational functions by the so‐called Froissart polynomial. In this paper the occurrence of the Froissart phenomenon is explored for the first time in a multivariate setting. Several obvious questions arise. Which definition of Padé approximant is to be used? Which multivariate rational functions should be investigated? When considering univariate projections of these functions, our analysis confirms the univariate results obtained so far in [13], under the condition that the noise is added after projection. At the same time, it is apparent from section 4 that for the unprojected multivariate Froissart polynomial no conjecture can be formulated yet.

     

Pade approximants as limits of rational functions of best approximation in Orlicz space

In this paper, we prove that the best rational approximation of a given analytic function in Orlicz space L^*(G), where G={⋎z⋎≤∈}, converges to the Padé approximants of the function as the measure of G approaches zero. This research is suported by National Science foundation Grant.     

Orthogonal rational functions and quadrature on the real half line

In this paper we generalize the notion of orthogonal Laurent polynomials to orthogonal rational functions. Orthogonality is considered with respect to a measure on the positive real line. From this, Gauss-type quadrature formulas are derived and multipoint Padé approximants for the Stieltjes transform of the measure. Convergence of both the quadrature formula and the multipoint Padé approximants is discussed.     

Equiconvergence in rational approximation of meromorphic functions

Generalizing the Walsh theorem, E. B. Saff, A. Sharma, and R. S. Varga showed that there is a close relation between the rational interpolants in roots of unity and Padé approximants of certain meromorphic functions. The purpose of this paper is to extend this result, replacing the Padé approximant with other rational functions so as to obtain a larger region of equiconvergence.     

The Best Multipoint Padé Approximant

Abstract

This paper is dealing with the problem of finding the “best” multipoint Padé approximant of an analytic function when data in some neighborhoods of sampling points are more important than others. More exactly, we obtain a multipoint Padé approximants as limits of best rational L^p-approximations on union of disks, when the measure of them tends to zero with different speeds. As such, this technique provides useful qualitative and analytic information concerning the approximants, which is difficult to obtain from a strictly numerical treatment.     

Asymptotic Two-Point Approximations by Rational Function

Continuous functions in 0 ≤ s ≤ ∞ with asymptotic power expansions both for t → +0 and t → +∞ are approximated by rational functions, being generalized Padé approximants both for s → +0 and s → +∞. The existence and the uniform convergence of such approximants are proved. As examples, piecewise rational functions, square roots of rational functions, and a modified error function are considered.     

Multivariate Padé-Bergman approximants

Summary. We define the multivariate Padé-Bergman approximants (also called Padé approximants) and prove a natural generalization of de Montessus de Ballore theorem. Numerous definitions of multivariate Padé approximants have already been introduced. Unfortunately, they all failed to generalize de Montessus de Ballore theorem: either spurious singularities appeared (like the homogeneous Padé [3,4], or no general convergence can be obtained due to the lack of consistency (like the equation lattice Padé type [3]). Recently a new definition based on a least squares approach shows its ability to obtain the desired convergence [6]. We improve this initial work in two directions. First, we propose to use Bergman spaces on polydiscs as a natural framework for stating the least squares problem. This simplifies some proofs and leads us to the multivariate Padé approximants. Second, we pay a great attention to the zero-set of multivariate polynomials in order to find weaker (although natural) hypothesis on the class of functions within the scope of our convergence theorem. For that, we use classical tools from both algebraic geometry (Nullstellensatz) and complex analysis (analytic sets, germs). Received December 4, 2001 / Revised version received January 2, 2002 / Published online April 17, 2002     

Magic Efficiency of the Approximation of Smooth Functions by Weighted Means of Two N-Point Padé Approximants

Ukrainian Mathematical Journal - We consider the approximation of smooth functions by two weighted N-point Padé approximants and present some numerical examples and the inequalities between...     

The compass (star) identity for vector-valued rational interpolants

The compass identity (Wynn's five point star identity) for Padé approximants connects neighbouring elements called N, S, E, W and C in the Padé table. Its form has been extended to the cases of rational interpolation of ordinary (scalar) data and interpolation of vector-valued data. In this paper, full specifications of the associated five point identity for the scalar denominator polynomials and the vector numerator polynomials of the vector-valued rational interpolants on real data points are given, as well as the related generalisations of Frobenius' identities. Unique minimal forms of the polynomials constituting the interpolants and results about unattainable points correspond closely to their counterparts in the scalar case. This revised version was published online in June 2006 with corrections to the Cover Date.     

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.