Linear operators that preserve some test functions

Let [a,b] be a compact interval of the real line IR. It is known that the classical theorem of Bohman-Korovkin states that in order that a sequence of positive linear operators (Lm), mapping into itself the space C[a,b] of continuous real-valued functions on [a,b], equipped with the uniform norm, to have the property that, for any f∈C[a,b], if m → ∞ we have lim Lmf = f, uniformly on [a,b], it is necessary and sufficient that such a convergence occur for a triplet of “test functions” from C[a,b], forming a so called Korovkin system. For C[a,b] the three monomials e0,e1,e2 where ej(x) := xj (j = 0,1,2), represent such a system.

In this paper, we define a new sequence of linear positive operators of integral type to approximate functions in the space,. First, we study the basic convergence theorem in simultaneous approximation and then study Voronovskaja-type asymptotic formula. Then, we estimate an error occurs by this approximation in the terms of the modulus of continuity. Next, we give numerical examples to approximate three test functions in the space by the sequence. Finally, we compare the results with the classical sequence of Szãsz operators  on the interval . It turns out that, the sequence  gives better than the results of the sequence  for the two test functions using in the numerical examples.

A “ K-Attainable” inequality related to the convergence of positive linear operators

An extension based on qR-integral for a sequence of operators

The paper aims at approximating functions through a sequence of linear positive operators of continuous type. First we define the Appell polynomials of dimension d(d≥2) and with their aid we expand the Szász–Mirakjan operators in the Durrmeyer sense. The investigation of the new class of operators involves the study of convergence by using the universal Bohman–Korovkin theorem and the indication of the error by using modulus of smoothness. The main results are given by the asymptotic expansion of both the operators and their derivatives with the explicit specification of all coefficients in a concise form. In particular, Voronovskaja type theorems are obtained.

This paper is interested in a new sequence of linear positive operators including degenerate Appell polynomials. We give a convergence theorem for these operators and obtain the quantitative estimation of the approximation by using modulus of continuity, Peetre’s 𝒦-functional, Lipschitz class functions and a Voronovskaja-type theorem. In addition, we give a Kantorovich modification of these operators and derive some approximation properties.

In this paper we prove asymptotic formulae for the sequences of linear positive operators on the space C2πα,β×R of the all bivariate continuous functions f:α,β×R→R which are 2π-periodic with respect to the second variable. Some examples are given.

Convergence of sequences of linear positive operators: Remarks and applications

The present paper deals with some convergence results concerning Gupta-type operators. Varied sequences of linear positive operators have been discussed and their approximation properties been studied in literature. This work is a collection of these operators introduced over the past three decades.

In the present paper we prove a Korovkin type approximation theorem for a sequence of positive linear operators acting from a weighted space Cρ1 into a weighted space Bρ2 with the use of a matrix summability method which includes both convergence and almost convergence. We also study the rates of convergence of these operators.

This paper presents an approximation method of integrable functions using a modified Barbosu operator, aimed at improving the rate of convergence in function approximation on the interval [0,1]. By introducing a suitable adjustment in the weight function, we construct a sequence of positive linear operators that better preserve the function's characteristics and demonstrate superior approximation behaviour, are studied. Theoretical error bounds are established in terms of the first and second order modulus of smoothness using their equivalence with the K-functionals. Finally, numerical experiments using test functions validate the theoretical findings and confirm that the King-type modified Barbosu operator achieve better approximation performance than the usual Kantorovich type Barbosu operator.

Some approximation results for Durrmeyer operators

In this paper, considering A-statistical convergence instead of Pringsheim’s sense for double sequences, we prove a Korovkin-type approximation theorem for sequences of positive linear operators defined on the space of all real valued Bögel-type continuous and periodic functions on the whole real two-dimensional space. A strong application is also presented. Furthermore, we obtain some rates of A-statistical convergence in our approximation.

In this paper, we introduce and study a new type of convergences using statistical convergence via the power series method and measurable convergence. We also study their relationship with other convergences. Further, we demonstrate Korovkin-type approximation theorems for double sequences of positive linear operators using these newly specified convergences, and we also provide illustrations that demonstrate how our proven theorems are better than their classical counterparts. Finally, we have determined rates of statistical product measurable convergence using the power series approach and the modulus of continuity.

Integral-type operators on continuous function spaces on the real line