Exact Order Of Approximation Research Articles

Approximation of complex q-Beta-Baskakov-Szász-Stancu operators in compact disk

Abstract The purpose this study is to present and investigate the q q -Beta-Baskakov-Szasz-Stancu operator. The operators are accompanied by Voronovskaja-type consequences, which include both an exact approximation order and a quantitative assessment, specifically within compact disks.

Approximation Theorems for Complex α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-Bernstein–Kantorovich Operators

In this paper, we introduce the complex form of α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-Bernstein–Kantorovich operators. Respectively, upper quantitative estimates for the complex α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-Bernstein–Kantorovich operator and its derivatives, Voronovskaya type result and the exact order of approximation of these operators are studied.

Exact approximation order and well-distributed sets

We prove that for a suitable Ahlfors regular metric measure space X and a function ψ:(0,∞)→(0,∞) from a suitable class of approximation functions, the Hausdorff dimensions of the set Wψ(Q) of all points ψ-well-approximable by a well-distributed subset Q⊂X, and the set Eψ(Q) of points that are exactly ψ-approximable by Q, coincide. This answers in a general setting, a question of Beresnevich-Dickinson-Velani in the case of approximation of reals by rationals, and answered by Bugeaud in that case using the continued-fraction expansion of reals. Our main result applies in particular to approximation by orbits of fixed points of a wide class of discrete groups of isometries acting on the boundary of hyperbolic metric spaces.

On complex modified Bernstein-Stancu operators

<p style='text-indent:20px;'>The present paper deals with complex form of a generalization of perturbed Bernstein-type operators. Quantitative upper estimates for simultaneous approximation, a qualitative Voronovskaja type result and the exact order of approximation by these operators attached to functions analytic in a disk centered at the origin with radius greater than 1 are obtained in this study.</p>

Fourier Dimension Estimates for Sets of Exact Approximation Order: The Well-Approximable Case

Abstract We obtain a Fourier dimension estimate for sets of exact approximation order introduced by Bugeaud for certain approximation functions $\psi $. This Fourier dimension estimate implies that these sets of exact approximation order contain normal numbers.

Approximation properties of complex genuine α–Bernstein‐Durrmeyer operators

Herein, we propose a complex form of genuine Bernstein‐Durrmeyer type operators depending on a non‐negative real parameter . We present the quantitative upper bound, Voronovskaja type result and the exact order of approximation for the derivatives of the operators attached to analytic functions on compact disks. These results validate the extension of approximation properties of complex genuine –Bernstein‐Durrmeyer type operators from real intervals to compact disks in the complex plane.

Sets of exact approximation order by complex rational numbers

For a nonincreasing function $\psi$, let $\textrm{Exact}(\psi)$ be the set of complex numbers that are approximable by complex rational numbers to order $\psi$ but to no better order. In this paper, we obtain the Hausdorff dimension and packing dimension of $\textrm{Exact}(\psi)$ when $\psi(x)=o(x^{-2})$. We also prove that the lower bound of the Hausdorff dimension is greater than $2-\tau/(1-2\tau)$ when $\tau=\limsup_{x\to\infty}\psi(x)x^2$ small enough.

A new generalization of complex Stancu operators

In this paper, we investigate approximation properties of the complex form of an extension of the Bernstein polynomials, defined by Stancu by means of a probabilistic method. We obtain quantitative upper estimates for simultaneous approximation and the exact order of approximation by these operators attached to analytic functions in closed disks. Also, we prove that the new generalized complex Stancu operators preserve the univalence, starlikeness, convexity, and spirallikeness in the unit disk.

Approximation and Geometric Properties of Complex $$\alpha $$ α -Bernstein Operator

In this paper, we consider the complex form of a new generalization of the Bernstein operator, depending on a non-negative real parameter. We obtain quantitative upper estimate for simultaneous approximation, a qualitative Voronovskaja type result and the exact order of approximation. Also, we present some shape preserving properties of the complex \(\alpha \)-Bernstein operator such as univalence, starlikeness, convexity and spirallikeness.

On Complex Modified Genuine Szász-Durrmeyer-Stancu Operators

In this paper, we introduce complex modified genuine Szasz-Durrmeyer-Stancu operators to improve the results obtained in [4] and present overconvergence properties of these operators. We obtain some estimates on the rate of convergence, a Voronovskaja-type result and the exact order of approximation for these operators attached to analytic functions of exponential growth on compact disks.

Approximation of functions by complex genuine Pólya-Durrmeyer operators

In this paper, the exact order of approximation and Voronovskaja type theorems with quantitative estimate for complex genuine Polya-Durrmeyer polynomials attached to analytic functions on compact disks are obtained. Our results show that extension of the complex genuine Polya-Durrmeyer polynomials from real intervals to compact disks in the complex plane extends approximation properties (with quantitative estimates).

Approximative properties of the summation methods of Fourier integrals

We study the order (velocity) of the approximation of functions on the axis by entire functions of exponential type not higher than σ as σ → ∞ (the linear and best approximations). The exact order of approximation of individual functions on ℝd by the classical summation methods of Fourier integrals (Gauss–Weierstrass, Bochner–Riesz, Marcinkiewicz) and the nonclassical Bernstein–Stechkin method is found. For functions on a torus, similar theorems of approximation by polynomials were obtained previously.

Approximation by Complex Baskakov-Szász-Durrmeyer Operators in Compact Disks

In this paper, we deal with the complex Baskakov-Szasz-Durrmeyer mixed operators and study Voronovskaja type results with quantitative estimates for these operators attached to analytic functions of exponential growth in the open disk of radius R. Also, the exact order of approximation is found. The method used allows to construct complex Szasz-type and Baskakov-type approximation operators without involving the values on the positive real axis.

Approximation by complex $$q$$ q -modified Bernstein–Schurer operators on compact disks

The purpose of this paper is to study the exact order of approximation and Voronovskaja type results with quantitative estimates for the complex \(q\)-modified Bernstein–Schurer operators \((0<q<1)\) attached to analytic functions on compact disks. In this way, we show the overconvergence phenomenon for these operators, namely the extensions of the approximation properties with quantitative estimates from the real intervals to compact disks in the complex plane.

Approximation by complex Szász-Mirakyan-Stancu-Durrmeyer operators in compact disks under exponential growth

In the present paper, we deal with the complex Sz?sz-Stancu-Durrmeyer operators and study Voronovskaja type results with quantitative estimates for these operators attached to analytic functions of exponential growth on compact disks. Also, the exact order of approximation is found.

Approximation by complex modified Szász-Mirakjan-Stancu operators in compact disks

In this paper, we establish some theorems on approximation and Voronovskaja type results for complex modified Sz?sz-Mirakjan-Stancu operators attached to analytic functions having exponential growth on compact disks. Also, we estimate the rate of convergence and the exact order of approximation.

Approximation by complex genuine hybrid operators

In the recent years several complex operators of summation-integral type have been proposed and their approximation properties have been discussed. The operators which have Baskakov or Szász basis functions have very complicated analysis. In the present article, we propose a complex hybrid operator, the special case of which provides the genuine Baskakov–Szász operators. Here we establish an upper estimate, Voronovskaja type asymptotic formula and exact order of approximation.

Approximation by a complex q-Baskakov-Stancu operator in compact disks

In this paper, we consider a complex q-Baskakov-Stancu operator and study some approximation properties. We give a quantitative estimate of the convergence, Voronovskaja-type result and exact order of approximation in compact disks.

Approximation by complex szász-durrmeyer operators in compact disks

In the present paper, we deal with the complex Szász-Durrmeyer operators and study Voronovskaja type results with quantitative estimates for these operators attached to analytic functions of exponential growth on compact disks. Also, the exact order of approximation is found.

Overconvergence of Complex Baskakov–Szász–Stancu Operators

In the present article, we study overconvergence of complex Baskakov–Szasz-Stancu operators and establish an upper estimate, a Voronovskaja asymptotic formula with quantitative estimates for these operators attached to analytic functions of exponential growth on compact disks. We also estimate the exact order of approximation. Our results essentially improve the results of the recent paper Gupta (Lobachevskii J Math 35(2):65–73, 2014).