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Constantin P. Niculescu

Publications and source records attributed to Constantin P. Niculescu.

At least 19 recordsLinked to original sources
arXiv2024

Functional inequalities in the framework of Banach spaces

A quadrilateral inequality established by C. Schötz in the context of Hilbert spaces is extended to the framework of Banach spaces. Our approach is based on the majorization theory and a substitute for the parallelogram law associated with Clarkson's notion of von Neumann-Jordan constant. As a by-product, several functional inequalities that extend classical inequalities from linear algebra and geometry of Banach spaces are also obtained.

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arXiv2023

Nonlinear operator extensions of Korovkin's theorems

In this paper we extend Korovkin's theorem to the context of sequences of weakly nonlinear and monotone operators defined on certain Banach function spaces. Several examples illustrating the theory are included.

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arXiv2023

The Hornich-Hlawka functional inequality for functions with positive differences

We analyze the role played by $n$-convexity for the fulfillment of a series of linear functional inequalities that extend the Hornich-Hlawka functional inequality, $f\left( x\right) +f\left( y\right) +f\left( z\right) +f\left( x+y+z\right) \geq f\left( x+y\right) +f\left( y+z\right)+f\left( z+x\right) +f(0),$ including extensions to the case of positive operators.

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arXiv2022

Quantitative Korovkin theorems for sublinear, monotone and strongly translatable operators in $L^{p}([0, 1]), 1\le p \le +\infty$

By extending the classical quantitative approximation results for positive and linear operators in $L^{p}([0, 1]), 1\le p \le +\infty$ of Berens and DeVore in 1978 and of Swetits and Wood in 1983 to the more general case of sublinear, monotone and strongly translatable operators, in this paper we obtain quantitative estimates in terms of the second order and third order moduli of smoothness, in Korovkin type theorems. Applications to concrete examples are included and an open question concerning interpolation theory for sublinear, monotone and strongly translatable operators is raised.

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arXiv2022

Korovkin type theorems for weakly nonlinear and monotone operators

In this paper we prove analogues of Korovkin's theorem in the context of weakly nonlinear and monotone operators acting on Banach lattices of functions of several variables. Our results concern the convergence almost everywhere, the convergence in measure and the convergence in $L^{p}$-norm. Several results illustrating the theory are also included.

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arXiv2021

A note on the Choquet type operators

In this note the Choquet type operators are introduced, in connection to Choquet's theory of integrability with respect to a not necessarily additive set function. Based on their properties, a quantitative estimate for the nonlinear Korovkin type approximation theorem associated to Bernstein-Kantorovich-Choquet operators is proved. The paper also includes a large generalization of Hölder's inequality within the framework of monotone and sublinear operators acting on spaces of continuous functions.

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arXiv2021

Nonlinear versions of Korovkin's abstract theorems

In this paper we prove Korovkin type theorems for sequences of sublinear, monotone and weak additive operators acting on function spaces C(X); where X is a compact or a locally compact metric space. Our results are illustrated by a series of examples

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arXiv2020

A note on the isotonic vector-valued convex functions

The property of isotonicity of a continuous convex function defined on the entire space or only on the positive cone is characterized via subdifferentials. Numerous examples illustrating the obtained results are included.

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arXiv2019

Kantorovich's Mass Transport Problem for Capacities

The aim of the present paper is to extend Kantorovich's mass transport problem to the framework of upper/lower continuous capacities and to prove the cyclic monotonicity of the supports of optimal supermodular plans. As in the probabilistic case, this easily yields the corresponding extension of the Kantorovich duality.

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arXiv2017

A note on Abel's partial summation formula

Several applications of Abel's partial summation formula to the convergence of series of positive vectors are presented. For example, when the norm of the ambient ordered Banach space is associated to a strong order unit, it is shown that the convergence of the series $\sum x_{n}$ implies the convergence in density of the sequence $(nx_{n})_{n}$ to 0. This is done by extending the Koopman-von Neumann characterization of convergence in density. Also included is a new proof of the Jensen-Steffensen inequality based on Abel's partial summation formula and a trace analogue of Tomić-Weyl inequality of submajorization.

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arXiv2014

Relative Convexity and Its Applications

We discuss a rather general condition under which the inequality of Jensen works for certain convex combinations of points not all in the domain of convexity of the function under attention. Based on this fact, an extension of the Hardy-Littlewood-Pólya theorem of majorization is proved and new insight is given into the problem of risk aversion in mathematical finance.

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