We study diagonal multipoint Padé approximants to sums of a Cauchy transform of a complex measure and a rational function. The measure is assumed to have compact regular support included into the real line and an argument of bounded variation on the support. For interpolation sets whose normalized counting measures converge sufficiently fast in the weak-star sense to some conjugate-symmetric distribution, we show that the counting measures of poles of the approximants converge to the balayage of that distribution onto the support of the measure, in the weak-star sense, that the approximants themselves converge in capacity to the approximated function outside the support of the measure, and that the poles of the additional rational function attract at least as many poles of the approximants as their multiplicity and not much more.
In this note we consider asymptotics of the multipoint Padé approximants to Cauchy integrals of analytic non-vanishing densities defined on a Jordan arc connecting $ -1 $ and $ 1 $. We allow for the situation where the (symmetric) contour attracting the poles of the approximants does separate the plane.
In this paper we investigate the question of uniform convergence of Padé approximants to elliptic functions that can be represented as Cauchy integrals of Dini-continuous non-vanishing densities given on 3-point Chebotarëv continua.
In this paper we study linear and non-linear Fourier-Padé approximation for Angelesco systems of functions. This construction is similar to that of Hermite-Padé approximation. Instead of considering power series expansions of the functions in the system, we take their expansion in a series of orthogonal polynomials.
In the present paper we prove a Stieltjes type theorem on the convergence of a sequence of rational functions associated with a mixed type Hermite-Padé approximation problem of a Nikishin system of functions and analyze the ratio asymptotic of the corresponding Hermite-Padé polynomials.
In this work type II Hermite-Padé approximants for a vector of Cauchy transforms of smooth Jacobi-type densities are considered. It is assumed that densities are supported on mutually disjoint intervals (an Angelesco system with complex weights). The formulae of strong asymptotics are derived for any ray sequence of multi-indices.
We consider Hermite-Padé approximants in the framework of discrete integrable systems defined on the lattice $\mathbb{Z}^2$. We show that the concept of multiple orthogonality is intimately related to the Lax representations for the entries of the nearest neighbor recurrence relations and it thus gives rise to a discrete integrable system. We show that the converse statement is also true. More precisely, given the discrete integrable system in question there exists a perfect system of two functions, i.e., a system for which the entire table of Hermite-Padé approximants exists. In addition, we …
The Padé table of $\phantom{}_2F_1(a,1;c;z)$ is normal for $c>a>0$ (cf. \cite{3}). For $m \geq n-1$ and $c \notin {\zz}^{\phantom{}^-}$, the denominator polynomial $Q_{mn}(z)$ in the $[m/n]$ Padé approximant $P_{mn}(z)/Q_{mn}(z)$ for $\phantom{}_2F_1(a,1;c;z)$ and the remainder term $Q_{mn}(z)\phantom{}_2F_1(a,1;c;z)-P_{mn}(z)$ were explicitly evaluated by Padé (cf. \cite{2}, \cite{5} or \cite{7}). We show that for $c>a>0$ and $m\geq n-1$, the poles of $P_{mn}(z)/Q_{mn}(z)$ lie on the cut $(1,\infty)$. We deduce that the sequence of approximants $P_{mn}(z)/Q_{mn}(z)$ converges to $\phantom{}_2…
Let $\widehatσ$ be a Cauchy transform of a possibly complex-valued Borel measure $σ$ and $\{p_n\}$ be a system of orthonormal polynomials with respect to a measure $μ$, $\mathrm{supp}(μ)\cap\mathrm{supp}(σ)=\varnothing$. An $(m,n)$-th Frobenius-Padé approximant to $\widehatσ$ is a rational function $P/Q$, $\mathrm{deg}(P)\leq m$, $\mathrm{deg}(Q)\leq n$, such that the first $m+n+1$ Fourier coefficients of the linear form $Q\widehatσ-P$ vanish when the form is developed into a series with respect to the polynomials $p_n$. We investigate the convergence of the Frobenius-Padé approximants to $\wi…
Given non-collinear points a_1, a_2, a_3, there is a unique compact, say Δ, that has minimal logarithmic capacity among all continua joining a_1, a_2, and a_3. For h be a complex-valued non-vanishing Dini-continuous function on Δ, we consider f_h(z) := (1/πi)\int_Δh(t)/(t-z) dt/w^+(t), where w(z) := \sqrt{\prod_{k=0}^3(z-a_k)} and w^+ the one-sided value according to some orientation of Δ. 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.
Let $ f_0 $ and $ f_\infty $ be formal power series at the origin and infinity, and $ P_n/Q_n $, with $ \mathrm{deg}(P_n),\mathrm{deg}(Q_n)\leq n $, be a rational function that simultaneously interpolates $ f_0 $ at the origin with order $ n $ and $ f_\infty $ at infinity with order $ n+1 $. When germs $ f_0,f_\infty $ represent multi-valued functions with finitely many branch points, it was shown by Buslaev that there exists a unique compact set $ F $ in the complement of which the approximants converge in capacity to the approximated functions. The set $ F $ might or might not separate the p…
First, an abstract scheme of constructing biorthogonal rational systems related to some interpolation problems is proposed. We also present a modification of the famous step-by-step process of solving the Nevanlinna-Pick problems for Nevanlinna functions. The process in question gives rise to three-term recurrence relations with coefficients depending on the spectral parameter. These relations can be rewritten in the matrix form by means of two Jacobi matrices. As a result, a convergence theorem for multipoint Padé approximants to Nevanlinna functions is proved.
We establish properties concerning the distribution of poles of Pad e approximants, which are generic in Baire category sense. We also investigate Pad e universal series, an analog of classical universal series, where Taylor partial sums are replaced with Pad e approximants. In particular, we complement previous studies on this subject by exhibiting dense or closed infi nite dimensional linear subspaces of analytic functions in a simply connected domain of the complex plane, containing the origin, whose all non zero elements are made of Pad e universal series. We also show how Pad e universal …
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.
Full indefinite Stieltjes moment problem is studied via the step-by-step Schur algorithm. Naturally associated with indefinite Stieltjes moment problem are generalized Stieltjes continued fraction and a system of difference equations, which, in turn, lead to factorization of resolvent matrices of indefinite Stieltjes moment problem. A criterion for such a problem to be indeterminate in terms of continued fraction is found and a complete description of its solutions is given in the indeterminate case. Explicit formulae for diagonal and sub-diagonal Padé approximants for formal power series corr…
Let f be a germ of an analytic function at infinity that can be analytically continued along any path in the complex plane deprived of a finite set of points, f \in\mathcal{A}(\bar{\C} \setminus A), \sharp A <\infty. J. Nuttall has put forward the important relation between the maximal domain of f where the function has a single-valued branch and the domain of convergence of the diagonal Pade approximants for f. The Pade approximants, which are rational functions and thus single-valued, approximate a holomorphic branch of f in the domain of their convergence. At the same time most of their pol…
We investigate the asymptotic behavior for type II Hermite-Pade approximation to two functions, where each function has two branch points and the pairs of branch points are separated. We give a classification of the cases such that the limiting counting measures for the poles of the Hermite-Pade approximants are described by an algebraic function of order 3 and genus 0. This situation gives rise to a vector-potential equilibrium problem for three measures and the poles of the common denominator are asymptotically distributed like one of these measures. We also work out the strong asymptotics f…
Convergence of diagonal Padé approximants is studied for a class of functions which admit the integral representation $ {\mathfrak F}(λ)=r_1(λ)\int_{-1}^1\frac{tdσ(t)}{t-λ}+r_2(λ), $ where $σ$ is a finite nonnegative measure on $[-1,1]$, $r_1$, $r_2$ are real rational functions bounded at $\infty$, and $r_1$ is nonnegative for real $λ$. Sufficient conditions for the convergence of a subsequence of diagonal Padé approximants of $ {\mathfrak F}$ on $\dR\setminus[-1,1]$ are found. Moreover, in the case when $r_1\equiv 1$, $r_2\equiv 0$ and $σ$ has a gap $(α,β)$ containing 0, it turns out that thi…
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 non-vanishing 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 orthogo…
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…
We study AAK as well as Padé approximants to functions f, where f is a sum of a Cauchy transform of a complex measure μsupported on a real interval included in (-1,1), whose Radon-Nikodym derivative with respect to the arcsine distribution on its support is Dini-continuous, non-vanishing and with and argument of bounded variation, and of a rational function with no poles on the support of μ. It is shown that the approximants converge to f locally uniformly in the domain of holomorphy of f, intersected with the unit disk in the case of AAK approximants. In the case of Padé approximants we need …
We solve a Padé-type problem of approximating three specific functions simultaneously by $q$-analogues of polylogarithms, respectively by powers of the logarithm. This problem is intimately related to recent results of the authors and Wadim Zudilin ["Séries hypergéométriques basiques, fonction $q$-zêta et séries d'Eisenstein", J. Inst. Math. Jussieu (to appear)] on the dimension of the vector space generated by $q$-analogues of values of the Riemann zeta function at integers. We also show that our result can be considered as a $q$-analogue of a result of Stéphane Fischler and the second author…
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…
Given $K$ arbitrary poles, which are neither necessarily distinct nor bounded, on the extended real line, a corresponding ordered base of rational functions orthogonal with respect to varying exponential weights is constructed: this gives rise to a $K$-fold family of orthogonal rational functions (ORFs). The ORF problem is characterised as a family of $K$ matrix Riemann-Hilbert problems (RHPs) on the extended real line, and a corresponding family of $K$ energy minimisation (variational) problems containing external fields with singular points is formulated, and the existence, uniqueness, and r…
