Statistical equi-equal convergence of positive linear operators

Positivity - Many researchers have been interested in the concept of statistical convergence because of the fact that it is stronger than the classical convergence. Also, the concepts of...     

Statistical approximation by positive linear operators on modular spaces

In this paper, we investigate the problem of statistical approximation to a function by means of positive linear operators defined on a modular space. Especially, in order to get more powerful results than the classical aspects we mainly use the concept of statistical convergence. A non-trivial application is also presented.     

Certain positive linear operators

Properties (including the approximating ones) are investigated of positive linear operators L_n(f; x) for which the relation $$L_n \left( {\left( {t - x} \right)f\left( t \right); x} \right) = \frac{{\varphi \left( x \right)}}{n}L'_n \left( {f\left( t \right); x} \right)$$ is fulfilled, as well as the properties of operators L_n ^(m)(f;x). The results are applicable, in particular, to Bernstein polynomials, to the operators of Mirak'yan, Baskakov, and others.     

On positive linear interpolation operators

В этой работе мы даем о бобщение понятия нор мальной системы точек, введен ного Фейером [3]. Наше определ ение включает и случа й бесконечного интерв ала (0, ∞). Доказано, в частности, что систе ма точек 0<x ₁ ⁽ⁿ⁾ /(n)<... _n ⁽ⁿ⁾ <∞ является нормальной в смысле нашего определения тогда и т олько тогда, когда вып олняются оценки — фиксированное чис ло, 0≦?<1. Мы доказываем, что есл и точкиx _k ⁽ⁿ⁾ /(n) являются ну лями многочлена ЛагерраL _n ^(α) (x), то они образуют норма льную систему в том и т олько том случае, когда ?1<α≦0. Мы получаем, таким обр азом, положительный интерполяционный пр оцесс для каждой нормальной системы т очек и устанавливаем теорему сходимости для того с лучая, когда эти точки являются ну лямиL _n ^(α) (x) при — 1相似文献   

Equi-statistical convergence of positive linear operators

Balcerzak, Dems and Komisarski [M. Balcerzak, K. Dems, A. Komisarski, Statistical convergence and ideal convergence for sequences of functions, J. Math. Anal. Appl. 328 (2007) 715-729] have recently introduced the notion of equi-statistical convergence which is stronger than the statistical uniform convergence. In this paper we study its use in the Korovkin-type approximation theory. Then, we construct an example such that our new approximation result works but its classical and statistical cases do not work. We also compute the rates of equi-statistical convergence of sequences of positive linear operators. Furthermore, we obtain a Voronovskaya-type theorem in the equi-statistical sense for a sequence of positive linear operators constructed by means of the Bernstein polynomials.     

Periodic points of positive linear operators and Perron-Frobenius operators

LetC(S) denote the Banach space of continuous, real-valued mapsf:S and letA denote a positive linear map ofC(S) into itself. We give necessary conditions that the operatorA have a strictly positive periodic point of minimal periodm. Under mild compactness conditions on the operatorA, we prove that these necessary conditions are also sufficient to guarantee existence of a strictly positive periodic point of minimal periodm. We study a class of Perron-Frobenius operators defined by

Unbounded functions and positive linear operators

The approximation of unbounded functions by positive linear operators under multiplier enlargement is investigated. It is shown that a very wide class of positive linear operators can be used to approximate functions with arbitrary growth on the real line. Estimates are given in terms of the usual quantities which appear in the Shisha-Mond theorem. Examples are provided.     

Existence and extensions of positive linear operators

In this paper we develop some unified methods, based on the technique of the auxiliary sublinear operator, for obtaining extensions of positive linear operators. In the first part, a version of the Mazur-Orlicz theorem for ordered vector spaces is presented and then this theorem is used in diverse applications: decomposition theorems for operators and functionals, minimax theory and extensions of positive linear operators. In the second part, we give a general sufficient condition (an implication between two inequalities) for the existence of a monotone sublinear operator and of a positive linear operator. Some particular cases in which this condition becomes necessary are also studied. Dedicated to Prof. Romulus Cristescu on his 80th birthday     

Lacunary equi-statistical convergence of positive linear operators

In this paper, the concept of lacunary equi-statistical convergence is introduced and it is shown that lacunary equi-statistical convergence lies between lacunary statistical pointwise and lacunary statistical uniform convergence. Inclusion relations between equi-statistical and lacunary equi-statistical convergence are investigated and it is proved that, under some conditions, lacunary equi-statistical convergence and equi-statistical convergence are equivalent to each other. A Korovkin type approximation theorem via lacunary equi-statistical convergence is proved. Moreover it is shown that our Korovkin type approximation theorem is a non-trivial extension of some well-known Korovkin type approximation theorems. Finally the rates of lacunary equi-statistical convergence by the help of modulus of continuity of positive linear operators are studied.      

Relative modular convergence of positive linear operators

In this paper we define a new type of modular convergence by using the notion of the relatively uniform convergence. We prove a Korovkin-type approximation theorem via this type of convergence in modular spaces. Then, we construct an example such that our new approximation result works but its classical cases do not work.     

On the iterates of positive linear operators

We have devised a new method for the study of the asymptotic behavior of the iterates of positive linear operators. This technique enlarges the class of operators for which the limit of the iterates can be computed.     

Statistical approximation theorems by k-positive linear operators

In this paper, using A-statistical convergence we obtain various approximation theorems by means of k-positive linear operators defined on the space of all analytic functions on the unit disk. Received: 17 February 2005     

Order of approximation by linear combinations of positive linear operators

Order of uniform approximation is studied for linear combinations due to May and Rathore of Baskakov-type operators and recent methods of Pethe. The order of approximation is estimated in terms of a higher-order modulus of continuity of the function being approximated.     

On infinite products of positive linear operators reproducing linear functions

Infinite products of positive linear operators (p.l.o.) reproducing linear functions are considered from a quantitative point of view. Refining and generalizing convergence theorems of Gwó?d?-?ukawska, Jachymski, Gavrea, Ivan and the present authors, it is shown that infinite products of certain positive linear operators, all taken from a finite set of mappings reproducing linear functions, weakly converge to the first Bernstein operator. A discussion of products of Meyer-König and Zeller operators is included.