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 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.     

Universal Padé approximants and their behaviour on the boundary

There are several kinds of universal Taylor series. In one such kind the universal approximation is required at every boundary point of the domain of definition \({\varOmega }\) of the universal function f. In another kind the universal approximation is not required at any point of \(\partial {\varOmega }\) but in this case the universal function f can be taken smooth on \(\overline{\varOmega }\) and, moreover, it can be approximated by its Taylor partial sums on every compact subset of \(\overline{\varOmega }\). Similar generic phenomena hold when the partial sums of the Taylor expansion of the universal function are replaced by some Padé approximants of it. In the present paper we show that in the case of Padé approximants, if \({\varOmega }\) is an open set and S, T are two subsets of \(\partial {\varOmega }\) that satisfy some conditions, then there exists a universal function \(f\in H({\varOmega })\) which is smooth on \({\varOmega }\cup S\) and has some Padé approximants that approximate f on each compact subset of \({\varOmega }\cup S\) and simultaneously obtain universal approximation on each compact subset of \((\mathbb {C}{\backslash }\overline{\varOmega })\cup T\). A sufficient condition for the above to happen is \(\overline{S}\cap \overline{T}=\emptyset \), while a necessary and sufficient condition is not known.     

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.     

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.     

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.     

The convergence of Padé approximants of meromorphic functions

It is proved that the [N, N + J] Padé approximants to any meromorphic function converge in measure within any bounded region of the complex plane as N approaches infinity.     

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.     

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.     

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.     

Convergence of simultaneous Padé approximants to two dilogarithms

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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.     

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)     

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.     

Solution of integral equations using function-valued Padé approximants II

We consider the use of functional (i.e. function-valued) Padé approximants to accelerate the convergence of Neumann series of linear integral equations and to estimate their characteristic values and eigenfunctions.We apply our methods to the Neumann series solution for the linear integral equation

Asymptotics of multipoint Hermite–Padé approximants of the first type for two beta functions

Mathematical Notes - The asymptotic behavior of the Hermite–Padé approximants of the first type for two beta functions are studied. The results are expressed in terms of equilibrium...