Steklov Means Research Articles

On the Steklov averages in operator cosine function framework

The Steklov averages (Steklov or integral means) are used in approximation theory of functions in different aspects. This article concerns the Steklov averages by using the operator cosine function framework. The operator cosine function offers a counterpart of the translation operator, which forms the basic concept for the modulus of continuity and for some approximation processes as well. We will show that the operator cosine function concept allows to define very general Steklov averages in an abstract Banach space. The approximation properties of these generalized Steklov averages appear to be quite similar to the properties of the Steklov averages in trigonometric approximation.

Estimates in direct inequalities for the Szász–Mirakyan operator

This paper deals with the approximation of continuous functions by the classical Szász–Mirakyan operator. We give new bounds for the constant in front of the second order Ditzian–Totik modulus of smoothness in direct inequalities. Asymptotic and non asymptotic results are stated. We use both analytical and probabilistic methods, the latter involving the representation of the operators in terms of the standard Poisson process. A smoothing technique based on a modification of the Steklov means is also applied.

On the rate of convergence of modified \(\alpha\)-Bernstein operators based on q-integers

In the present paper we define a q-analogue of the modified a-Bernstein operators introduced by Kajla and Acar (Ann. Funct. Anal. 10 (4) 2019, 570-582). We study uniform convergence theorem and the Voronovskaja type asymptotic theorem. We determine the estimate of error in the approximation by these operators by virtue of second order modulus of continuity via the approach of Steklov means and the technique of Peetre's \(K\)-functional. Next, we investigate the Gruss- Voronovskaya type theorem. Further, we define a bivariate tensor product of these operatos and derive the convergence estimates by utilizing the partial and total moduli of continuity. The approximation degree by means of Peetre's K- functional , the Voronovskaja and Gruss Voronovskaja type theorems are also investigated. Lastly, we construct the associated GBS (Generalized Boolean Sum) operator and examine its convergence behavior by virtue of the mixed modulus of smoothness.

Değişken üslü Lebesgue uzaylarında konvolüsyonların bazı özellikleri

A convolution in the variable exponent Lebesgue spaces is defined and the possibility its approximation by finite linear combinations of Steklov means is proved. Moreover, the convergence of the special convolutions sequence constructed via approximate identity to the original function is showed.

A study on convolutions in weighted lorentz spaces using steklov means

In this study, some estimations of convolution type operators defined with the help of Steklov operator in weighted Lorentz space L_ω^(p,q) (T) are obtained. Also, some basic properties of convolution type operators in these spaces are investigated.

Convergence of derivative of Szász type operators involving Charlier polynomials

<p style='text-indent:20px;'>The paper deals with the approximation of first order derivative of a function by the first order derivative of Szász-type operators based on Charlier polynomials introduced by Varma and Taşdelen [<xref ref-type="bibr" rid="b20">20</xref>]. The uniform convergence theorem, Voronovskaja type asymptotic theorem and an estimate of error in terms of the second order modulus of continuity of the derivative of the function are investigated. Further, it is shown that linear combinations of the derivative of the above operators converge to the derivative of function at a faster rate. Finally, an estimate of error in the approximation is obtained in terms of the <inline-formula><tex-math id="M1">\begin{document}$ (2k+2)th $\end{document}</tex-math></inline-formula> order modulus of continuity using Steklov mean.</p>

On integral generalization of Lupaş-Jain Operators

This paper mainly is a natural continuation of ?On Lupa?-Jain Operators? constructed by Ba?canbaz-Tunca et al. (Stud. Univ. Babe?-Bolyai Math. 63(4) (2018), 525-537) to approximate integrable functions on [0;1). We first present the weighted uniform approximation and provide a quantitative estimate for integral generalization of Lupa?-Jain operators. We also scrutinize the order of approximation in regards to local approximation results in sense of a classical approach, Peetre?s K-functional and Lipschitz class. Then, we prove that given operators can be approximated in terms of the Steklov means (Steklov averages). Lastly, a Voronovskaya-type asymptotic theorem is given.

Approximation Theorem for New Modification ofq-Bernstein Operators on (0,1)

In this work, we extend the works of F. Usta and construct new modifiedq-Bernstein operators using the second central moment of theq-Bernstein operators defined by G. M. Phillips. The moments and central moment computation formulas and their quantitative properties are discussed. Also, the Korovkin-type approximation theorem of these operators and the Voronovskaja-type asymptotic formula are investigated. Then, two local approximation theorems using Peetre’sK-functional and Steklov mean and in terms of modulus of smoothness are obtained. Finally, the rate of convergence by means of modulus of continuity and three different Lipschitz classes for these operators are studied, and some graphs and numerical examples are shown by using Matlab algorithms.

On Stancu-Type Generalization of Modified p , q -Szász-Mirakjan-Kantorovich Operators

In the present article, we construct p , q -Szász-Mirakjan-Kantorovich-Stancu operators with three parameters λ , α , β . First, the moments and central moments are estimated. Then, local approximation properties of these operators are established via K -functionals and Steklov mean in means of modulus of continuity. Also, a Voronovskaja-type theorem is presented. Finally, the pointwise estimates, rate of convergence, and weighted approximation of these operators are studied.

On Durrmeyer Typeλ-Bernstein Operators via (p,q)-Calculus

In the present paper, Durrmeyer typeλ-Bernstein operators via (p,q)-calculus are constructed, the first and second moments and central moments of these operators are estimated, a Korovkin type approximation theorem is established, and the estimates on the rate of convergence by using the modulus of continuity of second order and Steklov mean are studied, a convergence theorem for the Lipschitz continuous functions is also obtained. Finally, some numerical examples are given to show that these operators we defined converge faster in someλcases than Durrmeyer type (p,q)-Bernstein operators.

Approximation properties of modified q-Bernstein-Kantorovich operators

In the present paper we define a q-analogue of the modified Bernstein-Kantorovich operators introduced by Ozarslan and Duman (Numer. Funct. Anal. Optim. 37:92-105,2016). We establish the shape preserving properties of these operators e.g. monotonicity and convexity and study the rate of convergence by means of Lipschitz class and Peetre's K-functional and degree of approximation with the aid of a smoothing process e.g Steklov mean. Further, we introduce the bivariate case of modified q-Bernstein-Kantorovich operators and study the degree of approximation in terms of the partial and total modulus of continuity and Peetre's K-functional. Finally, we introduce the associated GBS (Generalized Boolean Sum) operators and investigate the approximation of the Bogel continuous and Bogel differentiable functions by using the mixed modulus of smoothness and Lipschitz class.

Inverse and Saturation Theorems for Some Summation Integral Linear Positive Operators

Inverse and saturation theorems have been studied for linear combination of some summation integral hybrid linear positive operators. Linear approximating technique and Steklov means have been used to obtain better approximation results.

Steklov means in Orlicz spaces

Steklov means in Orlicz spaces

Quantitative-Voronovskaya and Gr\xfcss-Voronovskaya type theorems for Sz\xe1sz-Durrmeyer type operators blended with multiple Appell polynomials

In this paper, we establish a link between the Szász-Durrmeyer type operators and multiple Appell polynomials. We study a quantitative-Voronovskaya type theorem in terms of weighted modulus of smoothness using sixth order central moment and Grüss-Voronovskaya type theorem. We also establish a local approximation theorem by means of the Steklov means in terms of the first and the second order modulus of continuity and Voronovskaya type asymtotic theorem. Further, we discuss the degree of approximation by means of the weighted spaces. Lastly, we find the rate of approximation of functions having a derivative of bounded variation.

On Some Properties of Convolutions in Variable Exponent Lebesgue Spaces

A convolution in the variable exponent Lebesgue spaces $$L_{2\pi }^{p\left( \cdot \right) }$$ is defined and its basic properties are investigated. It is also proved that this convolution can be approximated in $$L_{2\pi }^{p\left( \cdot \right) }$$ by the finite linear combinations of Steklov means of the original function.

Approximation of functions and their conjugates in variable Lebesgue spaces

One-sided Steklov means are used to introduce moduli of continuity of natural order in variable -spaces. A direct theorem of Jackson- Stechkin type and an inverse theorem of Salem-Stechkin type are given. Similar results are obtained for the conjugate functions. Bibliography: 24 titles.

Steklov averages as positive linear operators

We introduce a class of positive linear operators defined by Steklov means, investigate their properties and prove that the Weierstrass operators can be approximated in terms of the Steklov averages.

(p, q)-Szász–Mirakyan–Baskakov Operators

In the present paper we introduce (p, q) variant of Szasz–Mirakyan–Baskakov operators, we estimate moments and establish some approximation results which include weighted approximation and a direct estimate in terms of modulus of continuity. We have used linear approximating method namely Steklov mean.

Convergence and asymptotic stability of the explicit Steklov method for stochastic differential equations

In this paper, we develop a new numerical method with asymptotic stability properties for solving stochastic differential equations (SDEs). The foundations for the new solver are the Steklov mean and an exact discretization for the deterministic version of the SDEs. Strong consistency and convergence properties are demonstrated for the proposed method. Moreover, a rigorous linear and nonlinear asymptotic stability analysis is carried out for the multiplicative case in a mean-square sense and for the additive case in a path-wise sense using the pullback limit. In order to emphasize the characteristics of the Steklov discretization we use as benchmarks the stochastic logistic equation and the Langevin equation with a nonlinear potential of the Brownian dynamics. We show that the Steklov method has mild stability requirements and allows long-time simulations in several applications.

Lp --Approximation by a Linear Combination of Summation-integral Type Operators

The present paper is a study of some direct results in \(L_p\)−approximation by a linear combination of summation-integral type operators. We obtain an error estimate in terms of the higher order modulus of smoothness using some properties of the Steklov mean.