Extension properties for monotone and sublinear operators
In this paper we obtain several extension properties for monotone and sublinear operators. The results obtained generalize those known for positive and linear operators.
math.FA↗
arXiv subjects
Sorin G. Gal
Publications and source records attributed to Sorin G. Gal.
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.
math.FA↗
Quantitative estimates of convergence in nonlinear operator extensions of Korovkin's theorems
This paper is aimed to prove a quantitative estimate (in terms of the modulus of continuity) for the convergence in the nonlinear version of Korovkin's theorem for sequences of weakly nonlinear and monotone operators defined on spaces of continuous real functions. Several examples illustrating the theory are included.
math.FA↗
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.
math.FA↗
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.
math.FA↗
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.
math.CA↗
Choquet operators associated to vector capacities
The integral representation of Choquet operators defined on a space C(X) is established by using the Choquet-Bochner integral of a real-valued function with respect to a vector capacity.
math.FA↗
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
math.FA↗
Approximation of Random Functions by Random Polynomials in the Framework of Choquet's Theory of Integration
Given a submodular capacity space, we prove the uniform convergence in capacity and also the uniform convergence in the Choquet-mean of order $p\ge1$ with a quantitative estimate, of the multivariate Bernstein polynomials associated to a random function. Applications to quantitative estimates concerning the uniform convergence in capacity in the univariate case are given.
math.CA↗
Semi-discrete Gruss-Voronovskaya-type and Gruss-type estimates for Bernstein-Kantorovich polynomials
The aim of this note is to prove a semi-discrete Gruss-Voronovskaya-type estimate for Bernstein-Kantorovich polynomials. Also, as a consequence, a perturbed Gruss-type estimate is obtained.
math.CA↗
Volterra-Choquet nonlinear operators
In this paper we study to what extend some properties of the classical linear Volterra operators could be transferred to the nonlinear Volterra-Choquet operators, obtained by replacing the classical linear integral with respect to the Lebesgue measure, by the nonlinear Choquet integral with respect to a nonadditive set function. Compactness, Lipschitz and cyclicity properties are studied.
math.CA↗
A Nonlinear Extension of Korovkin's Theorem
In this paper we extend the classical Korovkin theorems to the framework of comonotone additive, sublinear and monotone operators. Based on the theory of Choquet capacities, several concrete examples illustrating our results are also discussed.
math.CA↗
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.
math.CA↗
Remarks on the Choquet Integral Calculus on $[a, t]$, with $a\in \mathbb{R}$
In this note, we extend the considerations for the Choquet integral calculus on the interval $[0, t]$ introduced in \cite{Su}, \cite{Su3}, to the case of an interval $[a, t]$, with arbitrary $a\in \mathbb{R}$.
math.GM↗
Polynomial approximation in slice regular Fock spaces
The main purpose of this paper is to prove some density results of polynomials in Fock spaces of slice regular functions. The spaces can be of two different kinds since they are equipped with different inner products and contain different functions. We treat both the cases, providing several results, some of them based on constructive methods which make use of the Taylor expansion and of the convolution polynomials. We also prove quantitative estimates in terms of higher order moduli of smoothness and in terms of the best approximation quantities.
math.CV↗
On the Gauss-Lucas theorem in the quaternionic setting
In theory of one complex variable, Gauss-Lucas Theorem states that the critical points of a non constant polynomial belong to the convex hull of the set of zeros of the polynomial. The exact analogue of this result cannot hold, in general, in the quaternionic case; instead, the critical points of a non constant polynomial belong to the convex hull of the set of zeros of the so-called symmetrization of the given polynomial. An incomplete proof of this statement was given in [8]. In this paper we present a different but complete proof of this theorem and we discuss a consequence.
math.CV↗
Approximation by convolutions with probability densities and applications to PDEs
The purpose of this paper is to introduce several new convolution operators, generated by some known probability densities. By using the inverse Fourier transform and taking inverse steps (in the analogues of the classical procedures used for, e.g., the heat or Laplace equations), we deduce the initial and final value problems satisfied by the new convolution integrals.
math.CA↗
On the Laws of Large Numbers in Possibility Theory
In this paper we obtain some possibilistic variants of the probabilistic laws of large numbers, different from those obtained by other authors, but very natural extensions of the corresponding ones in probability theory. Our results are based on the possibility measure and on the "maxitive" definitions for possibility expectation and possibility variance. Also, we show that in this frame, the weak form of the law of large numbers, implies the strong law of large numbers.
math.PR↗