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.      

Approximants of Śleszyński–Pringsheim continued fractions

We characterize the sequences {z_n} of complex numbers which are sequences of approximants of continued fractions K(a_n/b_n) with |a_n|+1⩽|b_n|, and study some of their properties. In particular we give truncation error bounds for such continued fractions.     

Direct and Inverse Results on Row Sequences of Hermite–Padé Approximants

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.     

Defects and the Convergence of Padé Approximants

A selective survey is given of convergence results for sequences of Padé approximants. Various approaches for dealing with the convergence problems due to `defects" are discussed. Attention is drawn to the close relationship between analyticity properties of a function and the `smoothness" of its Taylor series coefficients. A new theorem on the convergence of horizontal sequences of Padé approximants to functions in the Baker–Gammel–Wills conjecture function class is presented.     

Existence and Uniqueness of Matrix Padé Approximants

For the problems of the left and right matrix Padé approximations, we give the necessary and sufficient conditions for the existence of their solutions. If the left Padé approximant exists, then we prove that its uniqueness is equivalent to the existence of right Padé approximants, and we further give the exact results about the dimension of the linear space $^LR^{*}(m,n)$ formed from the left Padé approximants.     

Hermite—Padé Approximants of the Mittag-Leffler Functions

The convergence rate of type II Hermite–Padé approximants for a system of degenerate hypergeometric functions {₁F₁(1, γ; λ_jz)} _j=1 ^k is found in the case when the numbers {λ_j} _j=1 ^k are the roots of the equation λ^k = 1 or real numbers and \(\gamma\in\mathbb{C}\;\backslash\left\{0,-1,-2,...\right\}\). More general statements are obtained for approximants of this type (including nondiagonal ones) in the case of k = 2. The theorems proved in the paper complement and generalize the results obtained earlier by other authors.     

Convergence of Row Sequences of Simultaneous Padé–Faber Approximants

For meromorphic circumferentially mean p-valent functions, an analog of the classical distortion theorem is proved. It is shown that the existence of connected lemniscates of the function and a constraint on a cover of two given points lead to an inequality involving the Green energy of a discrete signedmeasure concentrated at the zeros of the given function and the absolute values of its derivatives at these zeros. This inequality is an equality for the superposition of a certain univalent function and an appropriate Zolotarev fraction.     

Convergence of Pad? Kernel Approximants to the Delayed Dirac Function

We derive from the solutions of Kummer's equation an expressionfor the exponential function, which when restricted to integerparameters gives the [M, N] Pad? approximation and its error.Laplace inversion, using Bromwich's integral, then yields anapproximation of the form to ( – t) the delayed Dirac kernel, in which the constants{_{i, v}, K_i,v: i = 1, 2 , ...,N, = M – N + 1} are thoseof the partial fraction decomposition of the [M, N] Pad? approximation.These constants also occur in direct quadrature formulae toinvert Laplace transforms. Finally, we show that the kernelapproximants converge for > t.     

Approximants de Padé-Hermite 2ème partie: programmation

Summary In this second paper we propose an algorithm for the construction of the -tables and of the Padé-Hermite tables [3]. Then we present some numerical applications.

     

Effective Conductivity of Nonlinear Two-Phase Media: Homogenization and Two-Point Padé Approximants

The aim of this paper is a study of the quasilinear transport equation, for instance the stationary heat equation. For periodically microheterogeneous media asymptotic homogenization has been performed with the local problem formulated as a minimization problem. The Golden–Papanicolaou integral representation theorem and some bounds developed for the linear equation have been extended. Two-point Padé approximants have been used to calculate bounds. Examples are also provided.     

Inequalities for Two-Point Padé Approximants to the Expansions of Stieltjes Functions in a Real Domain

General inequalities satisfied in real domain by two-point Padé approximants (2PPA) to two formal power expansions of Stieltjes functions have been derived. The obtained formulae: (i) generalize the existing inequalities for one-point Padé approximants, (ii) extend the validity of the inequalities reported in J. Comp. Appl. Math. 67 (1996), 59–72, from the particular case 0 k 2 to the general case 0 k , where (u,k) denotes a given number of coefficients of a power expansion of Stieltjes functions at infinity, (iii) clarify the properties of 2PPA estimations of the effective conductivity for regular composites.     

Approximants de Padé et séries hypergéométriques équilibrées

In this paper, we present and solve some very general new Padé approximant problems, whose solutions can be expressed with hypergeometric series. These series appear in the proofs of the irrationality of ζ(3), of infinitely many ζ(2n+1), and in essentially all results of this kind in the literature. We also prove two new Diophantine results with this method.     

A Note on Pad Approximants Pairs as Limits of Algebraic Polynomials Pairs of Weighted Best Approximation in Orlicz Spaces

In this short note, we show the behavior in Orlicz spaces of best approximations by algebraic polynomials pairs on union of neighborhoods, when the measure of them tends to zero.     

Track–Monitoring of Wheel–Rail–Systems

Inspection and maintenance of railway networks is a complex and expensive task. Special measurement vehicles are used to record the geometrical properties of railway lines within required time intervals. Due to the extent of measurement data the quality of railway track is evaluated considering only a few parameters. Although safety and comfort of wheel–rail–systems depend on the dynamical behavior, current inspection vehicles are not equipped to measure dynamic properties. In this paper, we will discuss a novel approach to evaluate the quality of railway tracks based on wheel–rail dynamics: Wheelset dynamics of subway trains are analyzed by Karhunen–Loève Transformation to extract the principal dynamics from the collected measurement data. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On theorems of Delsarte–McEliece and Chevalley–Warning–Ax–Katz

We present a theorem that generalizes the result of Delsarte and McEliece on the p-divisibilities of weights in abelian codes. Our result generalizes the Delsarte–McEliece theorem in the same sense that the theorem of N. M. Katz generalizes the theorem of Ax on the p-divisibilities of cardinalities of affine algebraic sets over finite fields. As the Delsarte–McEliece theorem implies the theorem of Ax, so our generalization implies that of N. M. Katz. The generalized theorem gives the p-divisibility of the t-wise Hamming weights of t-tuples of codewords (c ⁽¹⁾, . . . ,c ^(t)) as these words range over a product of abelian codes, where the t-wise Hamming weight is defined as the number of positions i in which the codewords do not simultaneously vanish, i.e., for which ${(c^{(1)}_i,\ldots,c^{(t)}_i)\not=(0,\ldots,0)}$ . We also present a version of the theorem that, for any list of t symbols s ₁, . . . ,s _t , gives p-adic estimates of the number of positions i such that ${(c^{(1)}_i,\ldots,c^{(t)}_i)=(s_1,\ldots,s_t)}$ as these words range over a product of abelian codes.     

Discrete potential Ablowitz–Kaup–Newell–Segur equation

We establish a discrete model for the potential Ablowitz–Kaup–Newell–Segur equation via a generalized Cauchy matrix approach. Soliton solutions and Jordan block solutions of this equation are presented by solving the determining equation set. By applying appropriate continuum limits, we obtain two semi-discrete potential Ablowitz–Kaup–Newell–Segur equations. The reductions to real and complex discrete and semi-discrete potential modified Korteweg-de Vries equations are also discussed.     

A Note on Pad ′e Approximants Pairs as Limits of Algebraic Polynomials Pairs of Weighted Best Approximation in Orlicz Spaces

In this short note, we show the behavior in Orlicz spaces of best approxima-tions by algebraic polynomials pairs on union of neighborhoods, when the measure of them tends to zero.     

On the Carey–Helton–Howe–Pincus trace formula

In this article, we give a new proof of the Carey–Helton–Howe–Pincus trace formula using Kato's theory of “relatively-smooth” operators and Krein's trace formula.