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Margareta Heilmann

Publications and source records attributed to Margareta Heilmann.

10 recordsLinked to original sources
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

Kernels for composition of positive linear operators

This paper investigates the composition of Bernstein--Durrmeyer operators and Sz\'asz--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 Sz\'asz--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.

math.CA↗
arXiv2022

Poisson approximation to the binomial distribution: extensions to the convergence of positive operators

The idea behind Poisson approximation to the binomial distribution was used in [J. de la Cal, F. Luquin, J. Approx. Theory, 68(3), 1992, 322-329] and subsequent papers in order to establish the convergence of suitable sequences of positive linear operators. The proofs in these papers are given using probabilistic methods. We use similar methods, but in analytic terms. In this way we recover some known results and establish several new ones. In particular, we enlarge the list of the limit operators and give characterizations of them.

math.PR↗
arXiv2018

A Nice Representation for a Link between Baskakov- and Szász-Mirakjan-Durrmeyer Operators and their Kantorovich Variants

In this paper we consider a link between Baskakov-Durrmeyer type operators and corresponding Kantorovich type modifications of their classical variants. We prove a useful representation for Kantorovich variants of arbitrary order which leads to a simple proof of convexity properties for the linking operators. This also solves an open problem. Another open problem is presented at the end of the paper.

math.CA↗
arXiv2012

On the composition and decomposition of positive linear operators III: A non-trivial decomposition of the Bernstein operator

The central problem in this technical report is the question if the classical Bernstein operator can be decomposed into nontrivial building blocks where one of the factors is the genuine Beta operator introduced by Mühlbach and Lupaş. We collect several properties of the Beta operator such as injectivity, the eigenstructure and the images of the monomials under its inverse. Moreover, we give a decomposition of the form $B_n = \bar{\mathbb{B}}_n \circ F_n $ where $F_n$ is a nonpositive linear operator having quite interesting properties. We study the images of the monomials under $F_n$, its moments and various representations. Also an asymptotic formula of Voronovskaya type for polynomials is given and a connection with a conjecture of Cooper and Waldron is established. In an appendix numerous examples illustrate the approximation behaviour of $F_n$ in comparison to $B_n$.

math.CA↗