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Autour du théorème de Roth

On Roth's theorem. The celebrated theorem of Roth, together with its generalizations given by Mahler and Ridout, gives a lower bound for the degree of approximation of one or more algebraic numbers with respect to a fixed set of valuations by elements of a fixed number field. An analogous result holds for function fields in characteristic zero. In this paper we do the following: (1) generalize Roth's theorem to the case of fields with a product formula in characteristic zero, removing any technical hypothesis from a previous result of Lang: (2) give a unified proof of Roth's theorem in the number field and function field cases; (3) provide a quantitative version of the general Roth's theorem, extending, even in the number field case, previous results of Bombieri and Van Der Poorten.

     

Asymptotic approximation by de la Vallée Poussin means and their derivatives

The concern of this paper is the study of local approximation properties of the de la Vallée Poussin means V_n. We derive the complete asymptotic expansion of the operators V_n and their derivatives as n tends to infinity. It turns out that the appropriate representation is a series of reciprocal factorials. All coefficients are calculated explicitly.     

Extension du domaine de définition d'une série de Dirichlet multiplicative

We prove a general theorem which allows to approach a multiplicative Dirichlet series with two variables by a known series.     

Grandes valeurs et nombres champions de la fonction arithmétique de Kalmár

The Kalmár function K(n) counts the factorizations n=x₁x₂…xr with xi?2(1?i?r). Its Dirichlet series is where ζ(s) denotes the Riemann ζ function. Let ρ=1.728… be the root greater than 1 of the equation ζ(s)=2. Improving on preceding results of Kalmár, Hille, Erd?s, Evans, and Klazar and Luca, we show that there exist two constants C₅ and C₆ such that, for all n, holds, while, for infinitely many n's, .An integer N is called a K-champion number if M<N⇒K(M)<K(N). Several properties of K-champion numbers are given, mainly about the size of the exponents and the number of prime factors in the standard factorization into primes of a large enough K-champion number.The proof of these results is based on the asymptotic formula of K(n) given by Evans, and on the solution of a problem of optimization.     

Local approximation results for Szász-Mirakjan type operators

In this study, motivating our earlier work [O. Duman and M.A. ?zarslan, Szász-Mirakjan type operators providing a better error estimation. Appl. Math. Lett. 20, 1184–1188 (2007)], we investigate the local approximation properties of Szász-Mirakjan type operators. The second modulus of smoothness and Petree’s K-functional are considered in proving our results. Received: 17 September 2007     

A shape-preserving quasi-interpolation operator satisfying quadratic polynomial reproduction property to scattered data

In this paper, we construct a univariate quasi-interpolation operator to non-uniformly distributed data by cubic multiquadric functions. This operator is practical, as it does not require derivatives of the being approximated function at endpoints. Furthermore, it possesses univariate quadratic polynomial reproduction property, strict convexity-preserving and shape-preserving of order 3 properties, and a higher convergence rate. Finally, some numerical experiments are shown to compare the approximation capacity of our quasi-interpolation operator with that of Wu and Schaback’s quasi-interpolation scheme.     

On integral bernstein operators in some classes of measurable bivariate functions

The two main theorems are concerned with the approximations of (complex-valued) functions on the real plane by sums of Bernstein pseudoentire functions. They are formulated and proved in Section 4, after prior determination of the suitable integral operators. Analogous results for pseudopolynomial approximations were obtained by Brudnyî, Gonska, and Jetter ([2],[3]).Research supported by KBN grant 2 1079 91 01.     

Statistical approximation theorems by k-positive linear operators

In this paper, using A-statistical convergence we obtain various approximation theorems by means of k-positive linear operators defined on the space of all analytic functions on the unit disk. Received: 17 February 2005     

Kolmogorov diameters of entire functions of given growth

The paper establishes the decrease rate of the Kolmogorov diameters of entire functions from the space L ²(0, ∞) in terms of minimal Blaschke products on the singularity sets of Borel transforms. Besides, in C(K) the decrease rate of the Kolmogorov diameters is calculated for entire functions with given finite order.     

Some metric properties on the GCF fraction expansion

In Schweiger (2003) [1], Fritz Schweiger introduced the algorithm of the generalized continued fraction (GCF), and in Zhong (2008) [2], T. Zhong studied some basic metric properties of the GCF. In this paper, under the restriction of −1<ε(k)?1, the “0-1” law and the central limit theorem of quotients in the GCF expansions are studied.     

Generalized Hermite-Padé approximation for Nikishin systems of three functions

Nikishin systems of three functions are considered. For such systems, the rate of convergence of simultaneous interpolating rational approximations with partially prescribed poles is studied. The solution is described in terms of the solution of a vector equilibrium problem in the presence of a vector external field.     

On a Sequence of Positive Linear Operators Associated with a Continuous Selection of Borel Measures

In this paper we introduce and study a new sequence of positive linear operators acting on the space of Lebesgue-integrable functions on the unit interval. These operators are defined by means of continuous selections of Borel measures and generalize the Kantorovich operators. We investigate their approximation properties by presenting several estimates of the rate of convergence by means of suitable moduli of smoothness. Some shape preserving properties are also shown. Dedicated to the memory of Professor Aldo Cossu     

An extension of a bound for functions in Sobolev spaces,with applications to (<Emphasis Type="Italic">m</Emphasis>, <Emphasis Type="Italic">s</Emphasis>)-spline interpolation and smoothing

Given a function f on a bounded open subset Ω of with a Lipschitz-continuous boundary, we obtain a Sobolev bound involving the values of f at finitely many points of . This result improves previous ones due to Narcowich et al. (Math Comp 74, 743–763, 2005), and Wendland and Rieger (Numer Math 101, 643–662, 2005). We then apply the Sobolev bound to derive error estimates for interpolating and smoothing (m, s)-splines. In the case of smoothing, noisy data as well as exact data are considered.     

Riemann-Hilbert problem associated with Angelesco systems

Angelesco systems of measures with Jacobi-type weights are considered. For such systems, strong asymptotics for the related multiple orthogonal polynomials are found as well as the Szeg?-type functions. In the procedure, an approach from the Riemann-Hilbert problem plays a fundamental role.     

Convergence rate of a new Bezier variant of Chlodowsky operators to bounded variation functions

In the present paper, we estimate the rate of pointwise convergence of the Bézier Variant of Chlodowsky operators _Cn,α$C_{n,\alpha }$ for functions, defined on the interval extending infinity, of bounded variation. To prove our main result, we have used some methods and techniques of probability theory.     

An Extension of Bernstein Polynomials on the Semi–Axis

The paper deals with the approximation of functions f on (0,+), where f can be singular at the origin, by means of Bernstein-type sequences. Error estimates in weighted uniform spaces with some converse results are given.Research was supported in part by grant INDAM-GNIM Progetto Equazioni Integrali e Problemi di Algebra Lineare Connessi.     

On the divergence of certain Hermite-Fejér interpolation

Summary In his paper [1]P. Turán discovers the interesting behaviour of Hermite-Fejér interpolation (based on the ebyev roots) not describing the derivative values at exceptional nodes {_n} _n=1 . Answering to his question we construct such exceptional node-sequence for which the mentioned process is bounded for bounded functions whenever –1<x<1 but does not converge for a suitable continuous function at any point of the whole interval [–1, 1].