L. G. González Ricardo, G. López Lagomasino, S. Medina Peralta
The present paper deals with the convergence properties of multi-level Hermite-Padé approximants for a class of meromorphic functions given by rational perturbations with real coefficients of a Nikishin system of functions, and study the zero location of the corresponding multiple orthogonal polynomials.
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We consider row sequences of vector valued Padé-Faber approximants (simultaneous Padé-Faber approximants) and prove a Montessus de Ballore type theorem.
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.
We study the convergence of type I Hermite-Padé approximation for a class of meromorphic functions obtained by adding a vector of rational functions with real coefficients to a Nikishin system of functions.
Rational Hermite-Padé Approximants vs Padé Approximants. Numerical Results
本文通过数值算例并结合自由 Van der Pol 方程频率函数的解析结构分析，展示基于 Hermite–Padé 多项式的有理逼近相较 Padé 逼近的优势。
We give a Montessus de Ballore type theorem for row sequences of Hermite-Padé approximations of vector valued analytic functions refining some results in this direction due to P.R. Graves-Morris and E.B. Saff. We do this introducing the notion of incomplete Padé approximation which contains, in particular, simultaneous Padé approximation and may be applied in the study of other systems of approximants as well.
Generic approximation of entire functions by their Padé approximants has been achieved in the past (\cite{3}). In the present article we obtain generic approximation of holomorphic functions on arbitrary open sets by sequences of their Padé approximants. Similar results hold with functions smooth on the boundary of their domain of definition. In addition, the approximation is valid simultaneously with respect to all centers of expansion.
Approximation of entire functions by their padé approximants has been examined in the past. It is true that generically such an approximation holds. However, examining this problem from another viewpoint, we obtain stronger generic results on functions defined on simply connected domains or even open sets of arbitrary connectivity.
The theory of universal Taylor series can be extended to the case of Padé approximants where the universal approximation is not realized by polynomials any more, but by rational functions, namely the Padé approximants of some power series. We present the first generic result in this direction, for Padé approximants corresponding to Taylor developments of holomorphic functions in simply connected domains. The universal approximation is required only on compact sets $K$ which lie outside the domain of definition and have connected complement. If the sets $K$ are additionally disjoint from the bo…
On Convergence of Rational Hermite-Padé Approximants
本文比较了Padé近似量与有理Hermite-Padé近似量在多值解析函数模型类中的收敛性质，证明在考虑的解析函数类中，有理Hermite-Padé近似量收敛速度更快。
In \cite{5} we proved that generically functions defined in any open set can be approximated by a sequense of their padé approximants, in the sense of uniform convergence on compacta. In this paper we examine a more particular space, $A^{\infty}(Ω)$, and prove that we can obtain similar approximation results with functions smooth on the boundary.
Starting from the orthogonal polynomial expansion of a function $F$ corresponding to a finite positive Borel measure with infinite compact support, we study the asymptotic behavior of certain associated rational functions (Padé-orthogonal approximants). We obtain both direct and inverse results relating the convergence of the poles of the approximants and the singularities of $F.$ Thereby, we obtain analogues of the theorems of E. Fabry, R. de Montessus de Ballore, V.I. Buslaev, and S.P. Suetin.
We study the convergence of sequences of type I and type II Hermite-Padé approximants for certain systems of meromorphic functions made up of rational modifications of Nikishin systems of functions.
In the paper, we propose two new conjectures about the convergence of Hermite Approximants of multivalued analytic functions of Laguerre class ${\mathscr L}$. The conjectures are based in part on the numerical experiments, made recently by the authors in [26] and [27].
Given a system of functions f = (f1, . . . , fd) analytic on a neighborhood of some compact subset E of the complex plane, we give necessary and sufficient conditions for the convergence with geometric rate of the common denominators of multipoint Hermite-Pade approximants. The exact rate of convergence of these denominators and of the approximants themselves is given in terms of the analytic properties of the system of functions. These results allow to detect the location of the poles of the system of functions which are in some sense closest to E.
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 $\OO$ of the universal function $f$. In another kind the universal approximation is not required at any point of $\partial\OO$ but in this case the universal function $f$ can be taken smooth on $\oO$ and, moreover, it can be approximated by it's Taylor partial sums on every compact subset of $\oO$. Similar generic phenomena hold when the partial sums of the Taylor expansion of the universal function are replaced by some Padé approximan…
Given a system of functions $\textup{\textbf{F}}=(F_1,\ldots,F_d),$ analytic on a neighborhood of some compact subset $E$ of the complex plane with simply connected complement, we define a sequence of vector rational functions with common denominator in terms of the orthogonal expansions of the components $F_k, k=1,\ldots,d,$ with respect to a sequence of orthonormal polynomials associated with a measure $μ$ whose support is contained in $E$. Such sequences of vector rational functions resemble row sequences of type II Hermite-Padé approximants. Under appropriate assumptions on $μ,$ we give ne…
We give necessary and sufficient conditions for the convergence with geometric rate of the denominators of linear Padé-orthogonal approximants corresponding to a measure supported on a general compact set in the complex plane. Thereby, we obtain an analogue of Gonchar's theorem on row sequences of Padé approximants.
Given a vector function ${\bf F}=(F_1,\ldots,F_d),$ analytic on a neighborhood of some compact subset $E$ of the complex plane with simply connected complement, we define a sequence of vector rational functions with common denominator in terms of the expansions of the components $F_k, k=1,\ldots,d,$ with respect to the sequence of Faber polynomials associated with $E$. Such sequences of vector rational functions are analogous to row sequences of type II Hermite-Padé approximation. We give necessary and sufficient conditions for the convergence with geometric rate of the common denominators of …
First we establish some generic universalities for Padé approximants in the closure $X^\infty(\OO)$ in $A^\infty(\OO)$ of all rational functions with poles off $\oO$, the closure taken in $\C$ of the domain $\OO\subset\C$.\ Next we give sufficient conditions on $\OO$ so that $X^\infty(\OO)=A^\infty(\OO)$.\ Some of these conditions imply that, even if the boundary $\partial\OO$ of a Jordan domain $\OO$ has infinite length, the integration operator on $\OO$ preserves $H^\infty(\OO)$ and $A(\OO)$ as well.\ We also give an example of a Jordan domain $\OO$ and a function $f\in A(\OO)$, such that it…
Given a regular compact set $E$ in the complex plane, a unit measure $μ$ supported by $\partial E,$ a triangular point set $β:= \{\{β_{n,k}\}_{k=1}^n\}_{n=1}^{\infty},β\subset \partial E$ and a function $f$, holomorphic on $E$, let $π_{n,m}^{β,f}$ be the associated multipoint $β-$ Padé approximant of order $(n,m)$. We show that if the sequence $π_{n,m}^{β,f}, n\inΛ, m-$ fixed, converges exact maximally to $f$, as $n\to\infty,n\inΛ$ inside the maximal domain of $m-$ meromorphic continuability of $f$ relatively to the measure $μ,$ then the points $β_{n,k}$ are uniformly distributed on $\partial …
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.
Not all limit points of poles of the Padé approximants are obstructions for poinwise convergence
本文通过构造反例证明，对于最后中间行的Padé逼近，其极点的极限点并非都是整行逐点收敛到可逼近函数的障碍。
Asymptotics for Hermite-Pade rational approximants for two analytic functions with separated pairs of branch points (case of genus 0)
研究了两个函数的第二类Hermite-Pade逼近的渐近性态，其中每个函数有两个分支点，且分支点对是分离的。我们给出了一个分类的情况下，厄米-帕德逼近的极点的极限计数措施所描述的一个代数函数的3阶和亏格0。这种情况下，产生了一个矢量势平衡问题的三个措施和极点的公分母渐近分布像这些措施之一。我们还利用一个刻画该Hermite-Pade逼近问题的3 × 3黎曼-希尔伯特问题，得到了相应的Hermite-Pade逼近的强渐近性.
For an arbitrary tuple of $m+1$ germs of analytic functions at a fixed point, we introduce the so-called polynomial Hermite-Padé $m$-system (of order $n$, $n\in\mathbb N$), which consists of $m$ tuples of polynomials; these tuples, which are indexed by a natural number $k\in[1,\dots,m]$, are called the $k$th polynomials of the Hermite-Padé $m$-system. We study the weak asymptotics of the polynomials of the Hermite-Padé $m$-system constructed at the point $\infty$ from the tuple of germs $[1, f_{1,\infty},\dotsc$, $f_{m,\infty}]$ of the functions $1, f_1,\dots,f_m$ that are meromorphic on some …
This paper investigates existence of the nonstandard Pade approximants introduced by Cherkaev and Zhang in J. Comp. Phys. 2009 for approximating the spectral function of composites from effective properties at different frequencies. The spectral functions contain microstructure information. Since this reconstruction problem is ill-posed [9], the well-performed Pade approach is noteworthy and requires further investigations. In this paper, we validate the assumption that the effective dielectric component of interest can be approximated by Pade approximants whose denominator has nonzero power o…
A remark on Chebyshev rational functions, multipoint Padé approximants and Noise
受多点Padé逼近在物理学界兴趣的启发，讨论切比雪夫有理函数如何产生多点Padé逼近，给出递推关系和连分数，并数值展示噪声对插值方案的影响。
A survey of direct and inverse type results for row sequences of Padé and Hermite-Padé approximation is given. A conjecture is posed on an inverse type result for type II Hermite-Padé approximation when it is known that the sequence of common denominators of the approximating vector rational functions has a limit. Some inverse type results are proved for the so called incomplete Padé approximants which may lead to the proof of the conjecture and the connection is discussed.
Hermite-Padé approximants of type II are vectors of rational functions with common denominator that interpolate a given vector of power series at infinity with maximal order. We are interested in the situation when the approximated vector is given by a pair of Cauchy transforms of smooth complex measures supported on the real line. The convergence properties of the approximants are rather well understood when the supports consist of two disjoint intervals (Angelesco systems) or two intervals that coincide under the condition that the ratio of the measures is a restriction of the Cauchy transfo…
Embedding AC Power Flow with Voltage Control in the Complex Plane : The Case of Analytic Continuation via Padé Approximants
本文提出将电压幅值约束的AC功率流问题嵌入复平面的方法，通过解析延拓理论解决传统数值方法的难题，展示Padé近似子在复函数延拓中的关键作用，并验证所提方法优于牛顿-拉夫森法和半正定松弛法。
The paper has two relatively distinct but connected goals; the first is to define the notion of Padé\ approximation of Weyl-Stiltjes transforms on an arbitrary compact Riemann surface of higher genus. The data consists of a contour in the Riemann surface and a measure on it, together with the additional datum of a local coordinate near a point and a divisor of degree $g$. The denominators of the resulting Padé--like approximation also satisfy an orthogonality relation and are sections of appropriate line bundles. A Riemann--Hilbert problem for a square matrix of rank two is shown to characteri…
We study the logarihtnmic asymptotic of multiple orthogonal polynomials arising in a mixed type Hermite-Padé approximation problem associated with the rational perturbation of a Nikishin system of functions. The formulas obtained allow to give exact estimates of the rate of convergence of the corresponding Hermite-Padé approximants.
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