Truncation,amplitude, and jitter errors on R for sampling series derivatives

In this paper, we investigate the error analysis of the derivative of the classical sampling theorem of bandlimited functions. We consider truncation, amplitude, and time-jitter errors. Both pointwise and uniform estimates are given. We derive analogues of the results of Piper (1975), Brown (1969), Jagerman (1966) and Li (1998) in a generalized manner. The amplitude and time-jitter errors are studied in the view of the works of Butzer (1983) and Butzer et al. (1988), provided that the bandlimited function satisfies some decay properties.     

A Unified Approach to Generalized Stirling Functions

Here presented is a unified approach to generalized Stirling functions by using generalized factorial functions, k-Gamma functions, generalized divided difference, and the unified expression of Stirling numbers defined in [16]. Previous well-known Stirling functions introduced by Butzer and Hauss [4], Butzer, Kilbas, and Trujilloet [6] and others are included as particular cases of our generalization. Some basic properties related to our general pattern such as their recursive relations, generating functions, and asymptotic properties are discussed,which extend the corresponding results about the Stirling numbers shown in [21] to the defined Stirling functions.     

A Survey on Recent Results of the Mathematical Seminar in Perugia, inspired by the Work of Professor P.L. Butzer

Here we give a survey on recent results in approximation theory obtained by a group of mathematicians of the University of Perugia, inspired by the work of Professor Butzer. We collect some results concerning linear integral operators in various settings.     

Rate of convergence for the Bézier variant of the Bleimann–Butzer–Hahn operators

In the present paper, the authors make use of some results in probability theory with a view to estimating the rate of convergence for the Bézier variant of the Bleimann–Butzer–Hahn operators for functions of bounded variation.     

Dedicated to Professor Paul L. Butzer for his 70th birthday.

Dedication

Dedicated to Professor Paul L. Butzer for his 70th birthday.     

Some condensation theorems for semigroup operators

In this paper we affirmatively answer an open question raised by P. L. Butzer and W. Dickmeis in [2] and give a new general theorem of the condensation of values for seminorm sequences. This continues our previous investigations [4].     

This issue is dedicated to Professor Paul L. Butzer on the occasion of his 70th Birthday

Dedication

This issue is dedicated to Professor Paul L. Butzer on the occasion of his 70th Birthday     

正线性算子对无穷区间上有界变差函数的点态逼近度

用ＢＶ［０，∞）表示在［０，∞）的每一有限子区间上为有界变差函数的函数构成的空间。用表示ＢＶ［０，∞）上的正线性算子，其中ｄｔＫｎ（ｘ，ｔ）是非负测度且，则有定理如果Ｌｎ（｜ｔ－ｘ｜β，ｘ）≤Ｃ（ｘ）／ｎｖ，，这里β＞０，ｖ≥１，Ｃ（ｘ）是一个与ｘ有关的常数，对ｆ∈ＢＶ［０，∞）和ｘ∈（０，∞）有这里作为应用，给出算子的逼近估计．     

ON TERM BY TERM DYADIC DIFFERENTIABILITY OF WALSH—KACZMARZ SERIES

The aim of this paper is to prove the following theorem concerning the term by term differentiation the-orem of Walsh-Kaczmarz series. Let (ck) be a decreasing real sequence withare integrable functions and f(x) is a. e. dyadic (or Butzer and Wagner) differentiate withThe function Kk means the kth Walsh-Kaczmarz function.     

Convexity Conditions for Generalized Riemann Derivable Functions

Several convexity theorems for generalized Riemann derivative are obtained. A generalized symmetric derivative is introduced which includes the usual symmetric Riemann derivative and with the help of this generalized symmetric derivative several other convexity theorems are established. A partial answer to a conjecture of Butzer and Kozakiewicz is given.     

Asymptotic expansion of multivariate Kantorovich type operators

In this paper we study the asymptotic expansion of sequences of multivariate Kantorovich type operators and their partial derivatives. In particular, we obtain the complete expansion for the Kantorovich Bernstein operators on the simplex and for two Kantorovich type modifications of the Bleimann, Butzer and Hahn operators that we introduce in the paper. AMS subject classification 41A36     

Local saturation of conservative operators

We state a result about the local saturation of sequences of linear operators that preserve the sign of the k-th derivative of the functions. We apply it to the well known approximation operators of Bernstein, Szász–Mirakjan, Meyer–König and Zeller, and Bleimann, Butzer and Hahn.     

A q-Analog of Newton’s Series,Stirling Functions and Eulerian Functions

— Recently, Butzer et al. [BH1, BH2, BHS] have studied some classical combinatorial functions such as factorial functions, Stirling numbers and Eulerian numbers of fractional orders. In the present paper we show that much the same is true in the case of the q-analogs. Meanwhile we give some results for the convergence of a q-Newton interpolation series.     

Linear Combinations of Gammaoperators in L p — Spaces

In 1951 P.L. Butzer defined in his thesis linear combinations of Bernstein polynomials. This type of combinations was extended to other operators by several authors. In this paper we define special linear combinations of Gammaoperators and we will prove a global direct result and an equivalence theorem. The proofs are based on a remarkable connection between the derivatives of the kernel of Gammaoperators and Laguerre polynomials and moreover between the moments of the Gammaoperators and Laguerre polynomials.     

Korovkin type theorem for non‐tensor Balázs type Bleimann,Butzer and Hahn operators

In this paper, we prove a certain Korovkin type approximation theorem by introducing new test functions. We introduce the non‐tensor Balázs type Bleimann, Butzer and Hahn operators and give the approximation property by using this new Korovkin theorem. Furthermore, we obtain the rate of convergence of these operators by means of modulus of continuity. Finally, we state the multivariate version of the abovementioned Korovkin type theorem. Copyright © 2014 John Wiley & Sons, Ltd.     

On the analytic extension of Stirling numbers of the first kind

We present an analytic extension of the unsigned Stirling numbers of the first kind that is in a certain sense unique in its coincidence with the Stirling polynomials. We examine and compare our extension to previous extensions of (signed) Stirling numbers of the first kind given by Butzer et al. (2007, J. Difference Equ. Appl., 13) and of the unsigned numbers given by Adamchik (1997, J. Comput. Appl. Math., 79). We also see a connection to the Riemann zeta function.     

A note on Weak-Type Interpolation and Function Spaces

This paper is concerned with an interpolation theorem for operatorsof generalized weak-type on rearrangement invariant spaces.As a tool, generalized average operators of D. W. Boyd, P. L.Butzer and F. Fehér, and L. Maligranda are employed.The theorem contains a result of D. W. Boyd [1] for Lebesguespaces and fills in a gap left by papers of E. M. Semenov [6]and M. Zippin [9].     

Statistical approximation properties of the Durrmeyer type q‐Bleimann,Butzer, and Hahn operators

In this paper, we introduce a Durrmeyer‐type generalization of q‐Bleimann, Butzer, and Hahn operators based on q‐integers and obtain statistical approximation properties of these operators with the help of the Korovkin type statistical approximation theorem. We also compute rates of statistical convergence of these q‐type operators by means of the modulus of continuity and Lipschitz‐type maximal function, respectively. Copyright © 2011 John Wiley & Sons, Ltd.     

Approximation properties of a generalization of positive linear operators

In the present paper we introduce a generalization of positive linear operators and obtain its Korovkin type approximation properties. The rates of convergence of this generalization is also obtained by means of modulus of continuity and Lipschitz type maximal functions. The second purpose of this paper is to obtain weighted approximation properties for the generalization of positive linear operators defined in this paper. Also we obtain a differential equation so that the second moment of our operators is a particular solution of it. Lastly, some Voronovskaja type asymptotic formulas are obtained for Meyer-König and Zeller type and Bleimann, Butzer and Hahn type operators.