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Ulrich Abel

Publications and source records attributed to Ulrich Abel.

16 recordsLinked to original sources
arXiv2026

Complete asymptotic expansion for a Durrmeyer variant of operators based on Hermite polynomials

In this paper, we study a Durrmeyer variant of the positive linear operators based on two-variable Hermite polynomials recently introduced by G. Krech (2016). Our main objective is to establish a complete asymptotic expansion for these operators as n tends to infinity for locally integrable functions of polynomial growth. The coefficients of the expansion are explicitly expressed in terms of the derivatives of the function. As a corollary, a Voronovskaja-type formula is obtained.

math.CA↗
arXiv2026

Korovkin type theorems for operators acting on functions of polynomial and exponential growth on $[0,\infty)$

We prove two Korovkin-type approximation theorems for sequences of positive linear operators acting on continuous functions on $[0,\infty)$. Under the assumption of pointwise convergence on suitable test functions, we establish pointwise convergence for all functions with polynomial or exponential growth. As direct applications, we obtain convergence results for the classical Baskakov and Sz\'asz--Mirakjan operators. The proposed method offers an elementary framework that can be applied to a broad class of positive linear operators.

math.NA↗
arXiv2025

A complete asymptotic expansion for the semi-exponential Post-Widder operators

In the present paper, we study the asymptotic properties of the semi-exponential Post-Widder operator. It is connected with $p(x) = x^2$. The main result is a pointwise complete asymptotic expansion valid for locally smooth functions of exponential growth. All coefficients are derived and explicitly given. As a special case we recover the complete asymptotic expansion for the classical Post-Widder operator.

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arXiv2025

Kernels for composition of positive linear operators

This paper investigates the composition of Bernstein--Durrmeyer operators and Szász--Mirakjan--Durrmeyer operators, focusing on the structure and properties of the associated kernel functions. In the case of the Bernstein--Durrmeyer operators, we establish new identities for the kernel arising from the composition of two and three operators, from which the commutativity of these operators follows naturally. Building on the eigenstructure of the Bernstein--Durrmeyer operator $M_n$, we obtain a representation of the iterate $M_n^r$ as a linear combination of the operators $M_k$, for $k=0,1,\dots,n$. We also address the composition of Szász--Mirakjan--Durrmeyer operators and revisit a known result giving an elementary proof.

math.CA↗
arXiv2024

Asymptotic properties for a general class of Szasz-Mirakjan-Durrmeyer operators

In this paper we introduce a general family of Szász--Mirakjan--Durrmeyer type operators depending on an integer parameter $j \in \mathbb{Z}$. They can be viewed as a generalization of the Szász--Mirakjan--Durrmeyer operators [9], Phillips operators [11] and corresponding Kantorovich modifications of higher order. For $j\in {\mathbb{N}}$, these operators possess the exceptional property to preserve constants and the monomial $x^{j}$. It turns out, that an extension of this family covers certain well-known operators studied before, so that the outcoming results could be unified. We present the complete asymptotic expansion for the sequence of these operators. All its coefficients are given in a concise form. In order to prove the expansions for the class of locally integrable functions of exponential growth on the positive half-axis, we derive a localization result which is interesting in itself.

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arXiv2023

Simultaneous approximation by operators of exponential type

There are many results on the simultaneous approximation by sequences of special positive linear operators. In the year 1978, Ismail and May as well as Volkov independently studied operators of exponential type covering the most classical approximation operators. In this paper we study asymptotic properties of these class of operators. We prove that under certain conditions, asymptotic expansions for sequences of operators belonging to a slightly larger class of operators, can be differentiated term-by-term. This general theorem contains several results which were previously obtained by several authors for concrete operators. One corollary states, that the complete asymptotic expansion for the Bernstein polynomials can be differentiated term-by-term. This implies a well-known result on the Voronovskaja formula obtained by Floater.

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arXiv2016

A problem of I. Raşa on Bernstein polynomials and convex functions

We present an elementary proof of a conjecture by I. Raşa which is an inequality involving Bernstein basis polynomials and convex functions. It was affirmed in positive very recently by the use of stochastic convex orderings. Moreover, we derive the corresponding results for Mirakyan-Favard-Szász operators and Baskakov operators.

math.CA↗
arXiv2016

New proofs of Melzak's identity

In their recent book on combinatorial identities, Quaintance and Gould devoted one chapter to Melzak's identity. We give new proofs for this identity and its generalization.

math.CO↗
arXiv2014

Complete Monotonicity and Zeros of Sums of Squared Baskakov Functions

We prove complete monotonicity of sums of squares of generalized Baskakov basis functions by deriving the corresponding results for hypergeometric functions. Moreover, in the central Baskakov case we study the distribution of the complex zeros for large values of a parameter. We finally discuss the extension of some results for sums of higher powers.

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arXiv2013

Geometric series of positive linear operators and inverse Voronovskaya theorem

We define the associated geometric series for a large class of positive linear operators and study the convergence of the series in the case of sequences of admissible operators. We obtain an inverse Voronovskaya theorem and we apply our results to the Bernstein operators, the Bernstein-Durrmeyer-type operators, and the symmetrical version of Meyer-König and Zeller operators.

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