Modulus Of Continuity Research Articles

On embedding into Lorentz spaces (a distant case)

In the work we study an upper bound for a non - increasing non - negative function in the space $L^{p}(0,1)$ by the modulus of continuity of a variable increment $\omega_{p,\alpha,\psi}(f,\delta)$. We show that for the increment of the function of form $f(x)-f(x+hx^{\alpha}\psi(x))$ in the bound the modulus of continuity casts into the form $\omega_{p,\alpha,\psi}(f,\frac{\delta}{\delta^{\alpha}\psi(\frac{1}{\delta})})$. We also study the embedding $\tilde H_{p,\alpha,\psi}^\omega \subset L(\mu,\nu)(\mu \not= \nu)$ (a distant case). We obtained necessary and sufficient conditions for the parameters $p$, $\alpha$, $\mu$, $\nu$ and the functions $\psi$, $\omega$ for this embedding.

Differentiable Hartman-Grobman Theorem via modulus of continuity: A sharp result on linearization in general Banach space

Differentiable Hartman-Grobman Theorem via modulus of continuity: A sharp result on linearization in general Banach space

On Simultaneous Approximation and Combinations of Lupas Type Operators

The purpose of the present paper is to study a sequence of linear and positive operators which was introduced by A. Lupas. First, we obtain estimate of moments of the operators and then prove a basic convergence theorem for simultaneous approximation. Further, we find error in approximation in terms of modulus of continuity of function. Finally, we establish a Voronovskaja asymptotic formula for linear combination of the above operators.

Approximation by double periodic functions in the class C^r_A

In this work we will consider approximation by polynomials in doubly periodic Weierstrass functions for functions that are analytic in a domain and continuous in its closure. This problem is closely related to the approximation by holomorphic polynomials in two variables of a function that is holomorphic in a domain on an elliptic curve. We assume that at the boundary of the region on the plane the length of the are is commensurable with length of the chord. This condition also applies to the region on the elliptic curve. The possibility of obtaining an approximation estimate that is consistent with the so-called the inverse theorem. i. e., with the restoration of the smoothness of the function by the speed of approximation, it was previously obtained for classed of function analytic in a domain for which in the closure of the domain the derivative of a given order has a modulus of continuity of the Holder type, with an order less than unity. The approximation method used earlier did not make it possible to study classes of analytic functions whose derivative of some order is limited. In this paper, we use a different approximation method to approximate by polynomials in doubly periodic Weierstrass functions the functions that are analytic in a domain and whose derivative of a given order is bounded in the domain.

On approximating properties of blended variant of Kantorovich-Stancu type Lupaş operators

We introduce blended variant of Kantorovich-Stancu type Lupa? operators and study convergence and q-statistical convergence properties using Korovkin theorem. We investigate rate of convergence interms of modulus of continuity, Peetre?s K-functional and Lipschitz functions. Some Direct results and Voronovskaja type theorem are established. Moreover, graphical analysis and error estimations are presented using MATLAB.

Generalization of Lupas-Kantorovich Operators Connected with Polya Distribution

The motive of this paper is to introduce the generalization of Lupaş-Kantorovich operators connected with Pólya distribution and establish the rate of convergence in terms of modulus of continuity. Furthermore, a Voronovskaja type asymptotic formula for these operators is studied. In the end, few numerical examples with graphical representation are added to depict the effect of convergence of the operators.

On the convergence properties of generalized Szász–Kantorovich type operators involving Frobenious–Euler–Simsek-type polynomials

<p>This work focuses on the study of approximation properties of functions by Szász type operators involving Frobenius–Euler–Simsek-type polynomials, which have become more popular recently because of their special characteristics and functional organization. The convergence properties such as uniformly convergence and pointwise convergence in terms of modulus of continuity and Peetre-$ \it K $ functional are investigated with the help of these sequences of operators in depth. This paper also includes the estimation of the error of the approximation of these sequences of operators to some particular class of functions. The estimates are depicted using the Maple scientific computing program and presented in tables.</p>

Stancu variant of Jakimovski-Leviatan-Durrmeyer operators involving Brenke type polynomials

<p style='text-indent:20px;'>Karaisa [<xref ref-type="bibr" rid="b29">29</xref>] presented Jakimovski- Leviatan- Durrmeyer type operators by means of Appell polynomials. In a similar manner, Wani et al. [<xref ref-type="bibr" rid="b43">43</xref>] proposed a sequence of Jakimovski-Leviatan-Durrmeyer type operators involving Brenke type polynomials which include Appell polynomials and Hermite polynomials. We note that the definitions of the operators given in both these papers are not correct. In the present article, we introduce a Stancu variant of the operators considered in [<xref ref-type="bibr" rid="b43">43</xref>] after correcting their definition. The definition of the operator proposed in [<xref ref-type="bibr" rid="b29">29</xref>] may be similarly corrected. We establish the Korovkin type approximation theorem and the rate of convergence by means of the usual modulus of continuity, Peetre's K-functional and the class of Lipschitz type functions for our operators. Next, we discuss the Voronovskaja and Gr<inline-formula><tex-math id="M1">\begin{document}$ \ddot{u} $\end{document}</tex-math></inline-formula>ss Voronovskaja type asymptotic theorems. Finally, we study the convergence of these operators in a weighted space and the Korovkin type weighted statistical approximation theorem.</p>

A new variant of the modified Bernstein-Kantorovich operators defined by Özarslan and Duman

In the present article, we introduce a new function $ \zeta $ in the modified Bernstein-Kantorovich operators defined by Özarslan and Duman (http://dx.doi.org/10.1080/01630563.2015.1079219) and properties of the function $ \zeta $ is define as: this is an infinitely differentiable function in the closed interval $ [0, 1] \; \text{such that}\; \zeta(0) = 0, \; \zeta(1) = 1 $ and $ \zeta'(\varkappa) $. We discussed an approximation properties with the help of the Bohman-Korovkin's type theorem and scrutinize the rate of convergence with the aid of modulus of continuity, Lipschitz type functions. Furthermore, we study the rate of convergence by means of derivatives of bounded variation and an approximation theorem by using of the Bohman-Korovkin's type theorem in $ \mathcal{A}- $Statistical convergence.In lastly, in order to validate our theoretical results, we provide some numerical experiments that are produced by a MATHEMATICA compiler and show that a careful choice of the function $ \zeta(\varkappa) $ for the both operators and we see that the our operator gives the better approximation results as compare to the modified Bernstein-Kantorovich operators defined by Özarslan (doi.org/10.1080/01630563.2015.1079219).

Approximation by q‐Post‐Widder Operators Based on a New Parameter

The purpose of this paper is to introduce q‐Post–Widder operators based on a new parameter and study their approximation properties. The moments and central moments are investigated. And some local approximation properties of these operators by means of modulus of continuity and Peetre’s K‐functional are presented. Furthermore, the rate of convergence for these operators is obtained. Weighted approximation and the quantitative q‐Voronovskaja type theorem are discussed. Finally, numerical illustrative examples have been given to show the convergence of these newly defined operators.

APPROXIMATION BY A COMPOSITION OF APOSTOL-GENOCCHI AND PALTANEA-DURRMEYER OPERATORS

The present paper deals with the Durrmeyer construction of operators based on a class of orthogonal polynomials called Apostol-Genocchi polynomials. For the proposed operators, we first establish a global approximation result followed by its convergence estimate in terms of usual, r-th and weighted modulus of continuity. We further study the asymptotic type results such as the Voronovskaya theorem and quantitative Voronovskaya theorem. Moreover, we estimate the rate of pointwise convergence of the proposed operators for functions of bounded variation defined on the interval (0,∞). Finally, the results are validated through graphical representations and an absolute error table.

Kantorovich Stancu Type Operator Including Generalized Brenke Polynomials

This article is concerned with the sequence of operators of Stancu’s-type, involving extended Brenke polynomials. We apply Korovkin’s theorem to the sequence of positive linear operators, discuss the uniform approximation of continuous functions on closed bounded intervals by known tools theory, and also consider the second modulus of continuity, Peetre’s K-functional and Lipschitz class, which are essential concepts in approximation theory.

Trigonometric generated Lp degree of approximation

In this article we continue the study of smooth Picard singular integral operators that started in [3], see there chapters 10-14. This time the foundation of our research is a trigonometric Taylor’s formula. We establish the Lp convergence of our operators to the unit operator with rates via Jackson type inequalities engaging the first Lp modulus of continuity. Of interest here is a residual appearing term. Note that our operators are not positive.

Parametric generalization of the modified Bernstein-Kantorovich operators

In the current article, a parametrization of the modified Bernstein-Kantorovich operators is studied. Then the Korovkin theorem, approximation properties and central moments of these operators are investigated. The rate of approximation of the operators is obtained by the help of modulus of continuity, functions from Lipschitz class and Peetre-$\mathcal{K}$ functional. Finally, some numerical examples are illustrated to show the effectiveness of the newly defined operators.

Coefficients of multiple Fourier-Haar series and variational modulus of continuity

In this paper, we introduce the concept of a variational modulus of continuity for functions of several variables, give an estimate for the sum of the coefficients of a multiple Fourier-Haar series in terms of the variational modulus of continuity, and prove theorems of absolute convergence of series composed of the coefficients of multiple Fourier-Haar series. In this paper, we study the issue of the absolute convergence for multiple series composed of the Fourier-Haar coefficients of functions of several variables of bounded p-variation. We estimate the coefficients of a multiple Fourier-Haar series in terms of the variational modulus of continuity and prove the sufficiency theorem for the condition for the absolute convergence of series composed of the Fourier-Haar coefficients of the considered function class. This paper researches the question: under what conditions, imposed on the variational modulus of continuity of the fractional order of several variables functions, there is the absolute convergence for series composed of the coefficients of multiple Fourier-Haar series.

Nagy type inequalities in metric measure spaces and some applications

We obtain a sharp Nagy type inequality in a metric space $(X,\rho)$ with measure $\mu$ that estimates the uniform norm of a function using its $\|\cdot\|_{H^\omega}$-norm determined by a modulus of continuity $\omega$, and a seminorm that is defined on a space of locally integrable functions. We consider charges $\nu$ that are defined on the set of $\mu$-measurable subsets of $X$ and are absolutely continuous with respect to $\mu$. Using the obtained Nagy type inequality, we prove a sharp Landau-Kolmogorov type inequality that estimates the uniform norm of a Radon-Nikodym derivative of a charge via a $\|\cdot\|_{H^\omega}$-norm of this derivative, and a seminorm defined on the space of such charges. We also prove a sharp inequality for a hypersingular integral operator. In the case $X={\mathbb R}_+^m\times {\mathbb R}^{d-m}$, $0\le m\le d$, we obtain inequalities that estimate the uniform norm of a mixed derivative of a function using the uniform norm of the function and the $\|\cdot\|_{H^\omega}$-norm of its mixed derivative.

Trigonometric Induced Multivariate Smooth Gauss–Weierstrass Singular Integrals Approximation

In this article, we employ the uniform and Lp, 1≤p<∞ approximation properties of general smooth multivariate singular integral operators over RN, N≥1. It is a trigonometric relief approach with detailed applications to the corresponding smooth multivariate Gauss–Weierstrass singular integral operators. The results are quantitative via Jackson-type inequalities involving the first uniform and Lp moduli of continuity.

Bivariate Bernstein Chlodovsky Operators Preserving Exponential Functions and Their Convergence Properties

This paper is devoted to an extension of the bivariate generalized Bernstein-Chlodovsky operators preserving the exponential function exp ( 2 , 2 ) where exp ( α , β ) = e − α x − β y , α , β ∈ R 0 + and x , y ≥ 0 . For these operators, we first examine the weighted approximation properties for continuous functions in the weighted space, and in the latter case, we also obtain the convergence rate for these operators using a weighted modulus of continuity. Then, we investigate the order of approximation regarding local approximation results via Peetre’s K-functional. We introduce the GBS (Generalized Boolean Sum) operators of generalized Bernstein-Chlodovsky operators, and we estimate the degree of approximation in terms of the Lipschitz class of Bögel continuous functions and the mixed modulus of smoothness. Finally, we provide some numerical and graphical examples with different values to demonstrate the rate of convergence of the constructed operators.

Fuzzy Korovkin type Theorems via deferred Cesaro and deferred Euler equi-statistical convergence

We establish a fuzzy Korovkin type approximation theorem by using \(eq-stat^{D}_{CE}\)(deferred Ces\'{a}ro and deferred Euler equi-statistical) convergence proposed by Saini et al. for continuous functions over \([a,b]\). Further, we determine the rate of convergence via fuzzy modulus of continuity.

SOME INEQUALITIES BETWEEN THE BEST SIMULTANEOUS APPROXIMATION AND MODULUS OF CONTINUITY IN THE WEIGHTED BERGMAN SPACE

Some inequalities between the best simultaneous approximation of functions and their intermediate derivatives, and the modulus of continuity in a weighted Bergman space are obtained. When the weight function is \(\gamma(\rho)=\rho^\alpha,\) \(\alpha>0\), some sharp inequalities between the best simultaneous approximation and an \(m\)th order modulus of continuity averaged with the given weight are proved. For a specific class of functions, the upper bound of the best simultaneous approximation in the space \(B_{2,\gamma_{1}},\) \(\gamma_{1}(\rho)=\rho^{\alpha},\) \(\alpha>0\), is found. Exact values of several \(n\)-widths are calculated for the classes of functions \(W_{p}^{(r)}(\omega_{m},q)\).