Ulrich Abel | Technische Hochschule Mittelhessen (original) (raw)

Papers by Ulrich Abel

Positivity, May 27, 2024

In this note we provide a new way to capture operators involving Laguerre polynomials by composit... more In this note we provide a new way to capture operators involving Laguerre polynomials by composition of an integral operator and a discrete operator. The new operator so obtained is a discrete operator. We give three examples by considering composition of Szász-Durrmeyer operator, exponential-type operator related to 2x 3/2 and the Phillips operator, respectively, with Szász-Mirakyan operators. In all cases we obtain positive linear, discretely defined operators which are based on Laguerre polynomials and approximate functions on the positive real half-axis.

The American Mathematical Monthly, 1986

MathLAB Journal, Apr 30, 2019

The identities of Choi, Lee, and Srivastava imply a formula proposed by Wilf. We show that these ... more The identities of Choi, Lee, and Srivastava imply a formula proposed by Wilf. We show that these identities are immediate consequences of the well-known product formulas for the sine function and the cosine function. Moreover, we prove a generalization.

Elemente der Mathematik, Feb 24, 2022

Acta Mathematica Hungarica, Aug 22, 2016

In this note we spotlight the linear and positive operators of discrete type {{{(R_n)}_{n\geqq1... more In this note we spotlight the linear and positive operators of discrete type {{{(R_n)}_{n\geqq1}}}$$(Rn)n≧1 known as Balázs–Szabados operators. We prove that this sequence enjoys the variation detracting property. The convergence in variation of {{{(R_{n}f)}_{n\geqq1}}}$$(Rnf)n≧1 to f is also proved. A generalization in Kantorovich sense is constructed and boundedness with respect to BV-norm is revealed.

Demonstratio Mathematica, 2003

In the present paper we introduce a Bézier variant of the Baskakov-Kantorovich operators and stud... more In the present paper we introduce a Bézier variant of the Baskakov-Kantorovich operators and study the rate of convergence for functions of bounded variation. Furthermore, we present the complete asymptotic expansion for the Baskakov-Kantorovich operators.

American Mathematical Monthly, Aug 1, 2003

... 4. B. Jacobson, On the mean value theorem for integrals, this MONTHLY 89 (1982) 300-301. ... ... more ... 4. B. Jacobson, On the mean value theorem for integrals, this MONTHLY 89 (1982) 300-301. ... Fachbereich MND, Fachhochschule Giessen-Friedberg, University of Applied Sciences, Wilhelm-Leuschner-Strafle 13, 61169 Friedberg, Germany Ulrich.Abel @ mnd.fh-friedberg.de ...

Journal of Approximation Theory, Oct 1, 2017

Abstract We present an elementary proof of a conjecture by I. Rasa which is an inequality involvi... more Abstract We present an elementary proof of a conjecture by I. Rasa 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–Szasz operators and Baskakov operators.

Mathematical Inequalities & Applications, 2019

Bennett [1] gave a generalization of Schur's theorem in order to study various momentpreserving t... more Bennett [1] gave a generalization of Schur's theorem in order to study various momentpreserving transformations. Recently, Su [5] confirmed a monotonicity conjecture of Bennett which is related to the generalized Schur's theorem and Haber's inequality. In this paper we present a short proof of this result which is based on a combinatorial identity. Moreover, we show that the function in Bennett's conjecture is not only monotonically decreasing but completely monotonic. Furthermore, we give its explicit representation as a Laplace integral which implies the complete monotonicity. Finally, we prove a multivariate version of the above-mentioned combinatorial identity.

Springer eBooks, Oct 7, 2022

College Mathematics Journal, Mar 1, 2015

Summary We give an elementary proof of a combinatorial identity involving powers of binomial coef... more Summary We give an elementary proof of a combinatorial identity involving powers of binomial coefficients.

arXiv (Cornell University), Mar 22, 2016

In their recent book [3] on combinatorial identities, Quaintance and Gould devoted one chapter [3... more In their recent book [3] on combinatorial identities, Quaintance and Gould devoted one chapter [3, Chapt. 7] to Melzak's identity. We give new proofs for this identity and its generalization.

Proceedings of the American Mathematical Society, Sep 4, 2018

We deduce a formula for the exact number of gridpoints (i.e., elements of Z d) in the extended d-... more We deduce a formula for the exact number of gridpoints (i.e., elements of Z d) in the extended d-dimensional cube nC d = [−n, +n] d on intersecting hyperplanes. In the special case of the hyperplanes {x ∈ R d | x 1 + • • • + x d = b}, b ∈ Z, these numbers can be written as a finite sum involving products of certain binomial coefficients. Furthermore, we consider the limit as n tends to infinity which can be expressed in terms of Euler-Frobenius numbers. Finally, we state a conjecture on the asymptotic behaviour of this limit as the dimension d tends to infinity.

Annales Polonici Mathematici, 2017

Approximation Theory and its Applications

... (22) (20) reveals the well-known fact that the operators Oa preserve all Note that, for there... more ... (22) (20) reveals the well-known fact that the operators Oa preserve all Note that, for there holds Since ... to Eq. (19) I--I q+l 9 <,> =2o,(,><S,x) E(r (O.af) (x) = io ' kl / ... Theory 67(1991), 284-- 302. Heilmann, M., L~-Saturation of Some Modified Bernstein Operators, J. Approx. ...

Proceedings of the American Mathematical Society, 2018

We deduce a formula for the exact number of gridpoints (i.e., elements of Z d \mathbb {Z}^{d} ) i... more We deduce a formula for the exact number of gridpoints (i.e., elements of Z d \mathbb {Z}^{d} ) in the extended d d -dimensional cube n C d = [ − n , + n ] d nC_{d}=\left [ -n,+n \right ] ^{d} on intersecting hyperplanes. In the special case of the hyperplanes { x ∈ R d ∣ x 1 + ⋯ + x d = b } \{ x\in \mathbb {R}^{d}\mid x_{1}+\cdots +x_{d} =b\} , b ∈ Z b\in \mathbb {Z} , these numbers can be written as a finite sum involving products of certain binomial coefficients. Furthermore, we consider the limit as n n tends to infinity which can be expressed in terms of Euler-Frobenius numbers. Finally, we state a conjecture on the asymptotic behaviour of this limit as the dimension d d tends to infinity.

The American Mathematical Monthly, 1985

Numerical Functional Analysis and Optimization, Jan 12, 2003

Abstract Recently, Müller introduced and studied left quasi interpolants of his Gamma operators. ... more Abstract Recently, Müller introduced and studied left quasi interpolants of his Gamma operators. In this article we present the complete asymptotic expansion of these operators. As a special case we obtain a Voronovskaja type theorem.

Applicable Analysis, Mar 1, 2011

We take a glance on the results concerning the Bleimann, Butzer and Hahn (BBH) operators obtained... more We take a glance on the results concerning the Bleimann, Butzer and Hahn (BBH) operators obtained in the period 1980-2009. Our list is not exhaustive nor exclusive. We apologise all authors possessing papers on the BBH operators and are not referred in this paper.