Lipschitz Class Research Articles

Approximation in quantum calculus of the Phillips operators by using the sequences of q-Appell polynomials

In this paper, we attempt to use the Dunkl analog to study the convergence properties of q-Phillips operators by using the q-Appell polynomials. By applying the new sequences of continuous functions νs,q(z)=(z−12[s]q)ϱ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\ u _{s,q}(z)=\\Bigl(z- \\frac{1}{2[s]_{q}}\\Bigr)_{\\varrho}$\\end{document} on [0,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$[0, \\infty )$\\end{document}, we construct an improved version of the q-Phillips operators. We calculate the qualitative outcomes in weighted Korovkin spaces to better understand the Phillips operators’ uniform convergence results. We obtain the approximation properties by use of the modulus of continuity and functions belonging to the Lipschitz class. Moreover, we give some direct theorems for the function belonging to Peetre’s K-functional.

Max-Product Sampling Kantorovich Operators: Quantitative Estimates in Functional Spaces

In this paper, we study the order of approximation for max-product sampling Kantorovich operators based upon generalized kernels in the setting of Orlicz spaces. We establish a quantitative estimate for the considered family of sampling-type operators using the Orlicz-type modulus of smoothness, which involves the modular functional of the space. From this result, it is possible to obtain the qualitative order of convergence when functions belonging to suitable Lipschitz classes are considered. On the other hand, in the compact case, we exploit a suitable definition of K-functional in Orlicz spaces in order to provide an upper bound for the approximation error of the involved operators. The treatment in the general framework of Orlicz spaces allows one to obtain a unifying theory on the rate of convergence, as the proved results can be deduced for a wide range of functional spaces, such as L p -spaces, interpolation spaces and exponential spaces.

Degree of approximation of functions conjugate to the functions belonging to Lip((𝜉1,𝜉2)𝑟)-class through double Nörlund means

Abstract In this paper, we determine the error of approximation of functions, conjugate to the functions of two variables ( 2 ⁢ π 2\pi -periodic in both variables) belonging to the Lipschitz class Lip ( ( ξ 1 , ξ 2 ) ; r \operatorname{Lip}((\xi_{1},\xi_{2});r ) ( r ≥ 1 r\geq 1 ), through the double Nörlund means of its conjugate double Fourier series. Additionally, in the form of corollaries, we present some particular cases of the theorem proved here.

Best Approximation and Inverse Results for Neural Network Operators

In the present paper we considered the problems of studying the best approximation order and inverse approximation theorems for families of neural network (NN) operators. Both the cases of classical and Kantorovich type NN operators have been considered. As a remarkable achievement, we provide a characterization of the well-known Lipschitz classes in terms of the order of approximation of the considered NN operators. The latter result has inspired a conjecture concerning the saturation order of the considered families of approximation operators. Finally, several noteworthy examples have been discussed in detail

Lipschitz and Dini–Lipschitz functions for the Sturm–Liouville transform

In this work, we introduce Lipschitz and Dini–Lipschitz classes of functions associated with the Sturm–Liouville operator L := d 2 d x 2 + A ′ ( x ) A ( x ) d d x , where A is a nonnegative function satisfying certain conditions; and we prove two versions of Titchmarsh's theorems for the Sturm–Liouville transform F . The harmonic analysis associated with the operator L plays an important role in establishing the results of this paper.

Parametric generalization of the [formula omitted]-Meyer–König–Zeller operators

In this paper, we introduce α parametric generalization of the Meyer–König–Zeller operators based on q-integer, which provides better error estimation then the usual q-MKZ operators. We obtain a differential recurrence relation satisfied by (α,q)-Meyer–König and Zeller operators and by using it, we calculate the moments t1−tm, m∈0,1,2,3,4 more efficiently. We compute the error of approximation by means of the modulus of continuity and modified Lipschitz class functionals. We further derive the Riccati differential equation satisfied by the first three moments. Graphical and numerical illustrative examples are also given to show the power of approximation with these new operators. Finally, an illustrative real-world example associated with the surface air temperature been investigated to demonstrate the modeling capabilities of these operators.

On approximation of functions, conjugate to the functions of several variables belonging to weighted Lipschitz and Zygmund classes

In the present paper, we introduce a new weighted Lipschitz class W(Lp(TN), ξ1(s1), . . . , ξN(sN)) and Zygmund class Z(Lp(TN), ξ1(s1), . . . , ξN(sN)) for N ∈ N, which generalizes the classes given in [7, 11]. We prove two theorems about
 the degree (error) of approximation of functions, conjugate to the N-variable functions (2π-periodic in each variable) belonging to these classes using the N-multiple matrix means of their N-multiple conjugate Fourier series. We improve the results of Móricz and Rhoades [6] and Móricz and Shi [7], which are given in the form of corollaries.

Optimal with Respect to Accuracy Recovery of Some Classes Functions by Fourier Series

Introduction. Function approximation (approximation or restoration) is widely used in data analysis, model building, and forecasting. The goal of function approximation is to find the function that best approximates the original function. This can be useful when the original function is too complex to analyze or when a model needs to be simplified for more efficient computation or interpretation. Function approximation is an important tool in science, engineering, economics, and other fields where data analysis and modeling are required. It allows you to simplify complex functions, identify patterns in the behavior of the object of study, and predict the value of a function beyond the available data. The purpose of the paper is consider the problems of approximation of a function, which on some interval is given by its values in some set of nodal points and belongs to some class of functions by trigonometric Fourier series with a given accuracy and at fulfillment of given constraints on its execution time. The main attention is paid to obtaining estimates of computational complexity (implementation time) and solving the problem of function approximation by Fourier series with a given or maximum possible accuracy using efficient algorithms for solving optimization problems. Results. The general formulation of the problem of approximation of functions by Fourier series in accordance with the technology of solving problems of computational and applied mathematics with specified values of quality characteristics is presented. Estimates of the error of the proposed approximation algorithms using for the computation of Fourier coefficients the optimal in accuracy and close to them quadrature formulas for the computation of integrals from rapidly oscillating functions of the classes of Helder and Lipschitz with given fixed values in the nodes of a fixed grid are given. The corresponding quadrature formulas and constructive estimates of the error of the method of approximation of functions of the specified classes are given. Estimates of computational complexity of the given algorithms are obtained, which allow us to set real constraints on the time of algorithm implementation with a given or maximum possible accuracy. Conclusions. A comprehensive analysis of the quality of the considered algorithms for the approximation of functions by Fourier series using the accuracy-optimal (or close to them) quadrature formulas for the computation of Fourier coefficients for the computation of integrals from rapidly oscillating functions is presented. The estimates of their main characteristics – accuracy and computational complexity – are obtained. Keywords: function approximation, Fourier series, Fourier series coefficients, approximation error, computational complexity.

A q‐Erkuş–Srivastava polynomials operator

Chan et al. (Integral Transforms Spec. Funct. 12 (2), 139–148, (2001)) constructed a multivariable extension of the Lagrange polynomials, popularly known as the Chan–Chyan–Srivastava polynomials. Altin and Erkuş (Integral Transforms Spec. Funct. 17, 239–244, (2006)) proposed Lagrange–Hermite polynomials in several variables. Erkuş and Srivastava (Integral Transforms Spec. Funct., 17, 267–273, (2006)) presented an unification (and generalization) of the Chan–Chyan–Srivastava polynomials and the multivariable Lagrange–Hermite polynomials, called as Erkuş–Srivastava polynomials. Duman (Taiwanese J. Math. 12 (2), 539‐543, (2008)) defined a ‐analogue of these generalized multivariable polynomials. Inspired by these studies, we construct a linear positive operator by means of the ‐Erkuş–Srivastava multivariable polynomials and study the Korovkin‐type theorems and the rate of convergence of these operators by using summability techniques of weighted ‐statistical convergence and the power series method. We also define a th‐order Taylor generalization of the multivariable polynomials operator and investigate the approximation of th‐order continuously differentiable Lipschitz class elements. Finally, we define the bivariate case of ‐Erkuş–Srivastava multivariable polynomials operator and study its ‐statistical convergence by using four‐dimensional matrix transformation.

Hardy decomposition of first order Lipschitz functions by Clifford algebra-valued harmonic functions

In this paper we solve the problem on finding a sectionally Clifford algebra-valued harmonic function, zero at infinity and satisfying certain boundary value condition related to higher order Lipschitz functions. Our main tool are the Hardy projections related to a singular integral operator arising in bimonogenic function theory, which turns out to be an involution operator on the first order Lipschitz classes. Our result generalizes the classical Hardy decomposition of Hölder continuous functions on a simple closed curve in the complex plane.

Titchmarsh and Boas-type theorems related to (κ,n)-Fourier transform

Abstract The aim of this paper is to prove a generalization of Titchmarsh’s theorems for the generalized Fourier transform called ( κ , n {\kappa,n} )-Fourier transform, where n is a positive integer and κ is a constant coming from Dunkl theory. As an application, we derive a ( κ , n ) {(\kappa,n)} -Fourier multiplier theorem for L 2 {L^{2}} Lipschitz spaces. Moreover, we give necessary conditions to ensure that f belongs to either one of the generalized Lipschitz classes of order m. This allows us to establish the analogue of the Boas-type result for ℱ κ , n {\mathcal{F}_{\kappa,n}} .

ДЕЯКІ АСИМПТОТИЧНІ ВЛАСТИВОСТІ РОЗВ’ЯЗКІВ ТРИГАРМОНІЙНИХ РІВНЯНЬ

The author considers the optimization problem for the triharmonic equation in the presence of specific boundary conditions. As a result, the triharmonic Poisson integral was constructed in Cartesian coordinates for the upper half-plane. The asymptotic properties of this operator on Lipschitz classes in a uniform metric were studied. An exact equality was found for the upper bound of the deviation of the Lipschitz class functions from the triharmonic Poisson integral defined in Cartesian coordinates for the upper half-plane in the metric space. The results obtained in the article demonstrate the connection between the methods of approximation theory and the principles of optimal decision theory. Keywords: optimization problem, class of Lipschitz functions, uniform metric, triharmonic Poisson integral.

Estimation of approximation error of a function having its derivatives belonging to lipschitz class of order - α by extended legendre wavelet (ELW) method and its applications

Estimation of approximation error of a function having its derivatives belonging to lipschitz class of order - α by extended legendre wavelet (ELW) method and its applications

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.

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.

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.

On the Newton-Broyden method for solving systems of nonlinear equations

The Newton-Broyden method for solving systems of nonlinear equations with operator decomposition is studied in this paper. A local convergence analysis is provided under classical Lipschitz conditions and results of numerical experiments are presented. The effectiveness of this method in solving various types of nonlinear systems is demonstrated.

On a third-order modulus of smoothness

In this paper we define a new third-order modulus of smoothness and we prove the conservation by Stancu operators of the Lipschitz classes defined with this modulus.

Boas-type results for Mellin transform

In this paper, we give necessary and sufficient conditions for a continuous on R + function f to belong the symmetric generalized Lipschitz classes defined by the Mellin translation in terms of growth or decay of some integrals connected with the Mellin transform. The results are counterparts of ones for classical Lipschitz classes and the classical (complex) Fourier transform obtained earlier by the author.

Some Approximation Properties of Operators Including Degenerate Appell Polynomials

ABSTRACT 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.