Ioan Raşa

Hyers–Ulam stability of the linear differential operator with nonconstant coefficients

Abstract We obtain some stability results for the linear differential operator of order one in Banach spaces. As a consequence we derive a result on Hyers–Ulam stability for the linear differential operator of higher order with nonconstant coefficients.

Full length article: The Fréchet functional equation with application to the stability of certain operators

We present a new approach to the classical Frechet functional equation. The results are applied to the study of Hyers-Ulam stability of Bernstein-Schnabl operators.

Kantorovich Operators of Order k

This article is concerned with the k-th order Kantorovich modification of the classical Bernstein operators B n , namely, , where D k f is the derivative of order k and I k f is an antiderivative of order k of the function f. These operators are most useful in simultaneous approximation. We give detailed expressions of the moments up to order four of and estimates of ratios of moments. Next, they are used to refine Voronovskayas result for derivative approximation by B n . The properties of as approximation operators are also investigated.

On Bernstein–Schnabl Operators on the Unit Interval

In this paper we study the Bernstein-Schnabl operators associated with a continuous selection of Borel measures on the unit interval. We investigate their approximation properties by presenting several estimates of the rate of convergence in terms of suitable moduli of smoothness. We also study some shape preserving properties as well as the preservation of the convexity. Moreover we show that their iterates converge to a Markov semigroup whose generator is a degenerate second order elliptic differential operator on the unit interval. Qualitative properties of this semigroup are also investigated together with its asymptotic behaviour.

Semigroups Associated to Mache Operators (II)

This short note contains some supplementary results concerning the operators introduced by D.H. Mache and the semigroup associated with them. Special attention is paid to the action of the operators and the semigroup on monomials.

On Some Classes of Diffusion Equations and Related Approximation Problems

Of concern is a class of second-order differential operators on the unit interval. The C0-semigroup generated by them is approximated by iterates of positive linear operators that are introduced here as a modification of Bernstein operators. Finally, the corresponding stochastic differential equations are also investigated, leading, in particular to the evaluation of the asymptotic behaviour of the semigroup.

Entropies and Heun functions associated with positive linear operators

We consider a parameterized probability distribution p ( x ) = ( p 0 ( x ) , p 1 ( x ) , ? ) and denote by S(x) the squared l2-norm of p(x). The properties of S(x) are useful in studying the Renyi entropy, the Tsallis entropy, and the positive linear operator associated with p(x). We show that for a family of distributions (including the binomial and the negative binomial distributions), S(x) is a Heun function reducible to the Gauss hypergeometric function 2F1. Several properties of S(x) are derived, including integral representations and upper bounds. Examples and applications are given, concerning classical positive linear operators.

Feller Semigroups, Bernstein type Operators and Generalized Convexity Associated with Positive Projections

We study the majorizing approximation properties of both Bernstein type operators and the corresponding Feller semigroups associated with a positive projection acting on the space of all continuous functions defined on a convex compact set.

Voronovskaya Type Theorems in Weighted Spaces

ABSTRACT In this article, we introduce a generalization of Gamma operators based on a function ρ having some properties and prove quantitative Voronovskaya and quantitative Grüss type Voronovskaya theorems via weighted modulus of continuity.

A generalization of Kantorovich operators for convex compact subsets

In this paper we introduce and study a new sequence of positive linear operators acting on function spaces defined on a convex compact subset. Their construction depends on a given Markov operator, a positive real number and a sequence of probability Borel measures. By considering special cases of these parameters for particular convex compact subsets we obtain the classical Kantorovich operators defined in the one-dimensional and multidimensional setting together with several of their wide-ranging generalizations scattered in the literature. We investigate the approximation properties of these operators by also providing several estimates of the rate of convergence. Finally, the preservation of Lipschitz-continuity as well as of convexity are discussed