Nikishin Systems Are Perfect

K. Mahler introduced the concept of perfect systems in the general theory he developed for the simultaneous Hermite–Padé approximation of analytic functions. We prove that Nikishin systems are perfect, providing by far the largest class of systems of functions for which this important property holds. As consequences, in the context of Nikishin systems, we obtain: an extension of Markov’s theorem to simultaneous Hermite–Padé approximation, a general result on the convergence of simultaneous quadrature rules of Gauss–Jacobi type, the logarithmic asymptotics of general sequences of multiple orthogonal polynomials, and an extension of the Denisov–Rakhmanov theorem for the ratio asymptotics of mixed type multiple orthogonal polynomials.     

Higher order recurrences and row sequences of Hermite–Padé approximation

ABSTRACT

We obtain extensions of the Poincaré and Perron theorems for higher order recurrence relations and apply them to obtain an inverse-type theorem for row sequences of the (type II) Hermite–Padé approximation of a vector of formal power series.     

Multipoint Hermite—Padé approximations for beta functions

We construct multipoint Hermite—Padé approximations for two beta functions generating the Nikishin system with infinite discrete measures and unbounded supports. The asymptotic behavior of the approximants is studied. The result is interpreted in terms of the vector equilibrium problem in logarithmic potential theory in the presence of an external field and constraints on measure.     

Hermite‐Padé Approximants for a Pair of Cauchy Transforms with Overlapping Symmetric Supports

Hermite‐Padé approximants of type II are vectors of rational functions with a 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 transform of a third measure (Nikishin systems). In this work we consider the case where the supports form two overlapping intervals (in a symmetric way) and the ratio of the measures extends to a holomorphic function in a region that depends on the size of the overlap. We derive Szeg?‐type formulae for the asymptotics of the approximants, identify the convergence and divergence domains (the divergence domains appear for Angelesco systems but are not present for Nikishin systems), and show the presence of overinterpolation (a feature peculiar for Nikishin systems but not for Angelesco systems). Our analysis is based on a Riemann‐Hilbert problem for multiple orthogonal polynomials (the common denominator).© 2016 Wiley Periodicals, Inc.     

On multiple orthogonal polynomials for three Meixner measures

Multiple orthogonal polynomials for three discrete Meixner measures with identical exponential decay at infinity are studied. These polynomials are the denominators of the type II Hermite–Padé approximants to some hypergeometric functions. The limit distribution of zeros of such polynomials scaled in a certain way is described in terms of equilibrium logarithmic potentials and in terms of algebraic curves.     

Fourier–Padé Approximants for Nikishin Systems

We study type I Fourier–Padé approximation for certain systems of functions formed by the Cauchy transform of finite Borel measures supported on bounded intervals of the real line. This construction is similar to type I Hermite–Padé approximation. Instead of power series expansions of the functions in the system, we take their development in a series of orthogonal polynomials. We give the exact rate of convergence of the corresponding approximants. The answer is expressed in terms of the extremal solution of an associated vector-valued equilibrium problem for the logarithmic potential.      

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...     

Inverse results on row sequences of Hermite–Padé approximation

We consider row sequences of (type II) Hermite–Padé approximations with common denominator associated with a vector f of formal power expansions about the origin. In terms of the asymptotic behavior of the sequence of common denominators, we describe some analytic properties of f and restate some conjectures corresponding to questions once posed by A. A. Gonchar for row sequences of Padé approximants.     

Essentially negative news about positive systems

In this paper the discretisation of switched and non-switched linear positive systems using Padé approximations is considered. Padé approximations to the matrix exponential are sometimes used by control engineers for discretising continuous time systems and for control system design. We observe that this method of approximation is not suited for the discretisation of positive dynamic systems, for two key reasons. First, certain types of Lyapunov stability are not, in general, preserved. Secondly, and more seriously, positivity need not be preserved, even when stability is. Finally we present an alternative approximation to the matrix exponential which preserves positivity, and linear and quadratic stability.     

Rational approximation of analytic functions

The paper provides an overview of the author’s contribution to the theory of constructive rational approximations of analytic functions. The results presented are related to the convergence theory of Padé approximants and of more general rational interpolation processes, which significantly expand the classical theory’s framework of continuous fractions, to inverse problems in the theory of Padé approximants, to the application of multipoint Padé approximants (solutions of Cauchy-Jacobi interpolation problem) in explorations connected with the rate of Chebyshev rational approximation of analytic functions and to the asymptotic properties of Padé-Hermite approximation for systems of Markov type functions.     

Equivalence of a Scalar and a Vector Equilibrium Problem for a Pair of Functions Forming a Nikishin System

Mathematical Notes - We prove the equivalence of a vector and a scalar equilibrium problem that naturally arise when studying the limit distribution of zeros of type I Hermite–Padé...     

Two Families of Approximation Schemes for Delay Systems

Based on the Trotter-Kato approximation theorem for strongly continuous semigroups we develop a general framework for the approximation of delay systems. Using this general framework we construct two families of concrete approximation schemes. Approximation of the state is done by functions which are piecewise polynomials on a mesh (m-th order splines of deficiency m). For the two families we also prove convergence of the adjoint semigroups and uniform exponential stability, properties which are essential for approximation of linear quadratic control problems involving delay systems. The characteristic matrix of the delay system is in both cases approximated by matrices of the same structure but with the exponential function replaced by approximations where Padé fractions in the main diagonal resp. in the diagonal below the main diagonal of the Padé table for the exponential function play an essential role.     

On parallel algorithms for semidiscretized parabolic partial differential equations based on subdiagonal Padé approximations

It is well known that methods for solving semidiscretized parabolic partial differential equations based on the second-order diagonal [1/1] Padé approximation (the Crank–Nicolson or trapezoidal method) can produce poor numerical results when a time discretization is imposed with steps that are “too large” relative to the spatial discretization. A monotonicity property is established for all diagonal Padé approximants from which it is shown that corresponding higher-order methods suffer a similar time step restriction as the [1/1] Padé. Next, various high-order methods based on subdiagonal Padé approximations are presented which, through a partial fraction expansion, are no more complicated to implement than the first-order implicit Euler method based on the [0/1] Padé approximation; moreover, the resulting algorithms are free of a time step restriction intrinsic to those based on diagonal Padé approximations. Numerical results confirm this when various test problems from the literature are implemented on a Multiple Instruction Multiple Data (MIMD) machine such as an Alliant FX/8. © 1993 John Wiley & Sons, Inc.     

Upper bounds for the moduli of zeros of Hermite–Padé approximations for a set of exponential functions

In this paper, we establish upper bounds for the moduli of zeros of Hermite–Padé approximations of type I for a system of exponential functions \(\left\{ {{e^{{\lambda _{{p^z}}}}}} \right\}_{p = 0}^k\), where \(\left\{ {{\lambda _p}} \right\}_{p = 0}^k\) are various arbitrary complex numbers. The proved statements supplement and generalize well-known results due to Saff and Varga, as well as those due to Stahl and Wielonsky, on the behavior of zeros of Hermite–Padé approximations for a set of exponential functions \(\left\{ {{e^{pz}}} \right\}_{p = 0}^k\).     

Analytical solution of three-dimensional Navier–Stokes equations for the flow near an infinite rotating disk

In this letter, we will consider variational iteration method (VIM) and Padé approximant, for finding analytical solutions of three-dimensional viscous flow near an infinite rotating disk. The solutions is compared with the numerical (fourth-order Runge–Kutta) solution. The results illustrate that VIM–Padé is an appropriate method in solving the systems of nonlinear equations. It is predicted that VIM–Padé can have wide application in engineering problems (especially for boundary-layer and natural convection problems).     

Convergence and computation of simultaneous rational quadrature formulas

We discuss the convergence and numerical evaluation of simultaneous quadrature formulas which are exact for rational functions. The problem consists in integrating a single function with respect to different measures using a common set of quadrature nodes. Given a multi-index n, the nodes of the integration rule are the zeros of the multi-orthogonal Hermite–Padé polynomial with respect to (S, α, n), where S is a collection of measures, and α is a polynomial which modifies the measures in S. The theory is based on the connection between Gauss-type simultaneous quadrature formulas of rational type and multipoint Hermite–Padé approximation. The numerical treatment relies on the technique of modifying the integrand by means of a change of variable when it has real poles close to the integration interval. The output of some tests show the power of this approach in comparison with other ones in use.     

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.

     

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.     

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.     

A look-ahead method for computing vector Padé-Hermite approximants

In this paper, we present an algorithm to compute vector Padé-Hermite approximants along a sequence of perfect points in the vector Padé-Hermite table. We show the connection to matrix Padé approximants. The algorithm is used to compute the solution of a block Hankel system of linear equations.