scholarly journals Some New Results on Bicomplex Bernstein Polynomials

Mathematics ◽  
2021 ◽  
Vol 9 (21) ◽  
pp. 2748
Author(s):  
Carlo Cattani ◽  
Çíğdem Atakut ◽  
Özge Özalp Güller ◽  
Seda Karateke
Keyword(s):  
Analytic Functions ◽  
Quantitative Estimate ◽  
Bernstein Polynomials ◽  
Approximation Properties ◽  
Type Result ◽  
Complex Bernstein Polynomials

The aim of this work is to consider bicomplex Bernstein polynomials attached to analytic functions on a compact C2-disk and to present some approximation properties extending known approximation results for the complex Bernstein polynomials. Furthermore, we obtain and present quantitative estimate inequalities and the Voronovskaja-type result for analytic functions by bicomplex Bernstein polynomials.

Voronovskaja’s theorem, shape preserving properties and iterations for complex q-Bernstein polynomials

2011 ◽  
Vol 48 (1) ◽  
pp. 23-43 ◽  
Author(s):  
Sorin Gal
Keyword(s):  
Convergence Theorem ◽  
Analytic Functions ◽  
Quantitative Estimate ◽  
Bernstein Polynomials ◽  
Exact Order ◽  
Approximation Properties ◽  
Shape Preserving ◽  
Voronovskaja’S Theorem ◽  
Shape Preserving Properties ◽  
Compact Disks

In this paper, first we prove Voronovskaja’s convergence theorem for complex q-Bernstein polynomials, 0 < q < 1, attached to analytic functions in compact disks in ℂ centered at origin, with quantitative estimate of this convergence. As an application, we obtain the exact order in approximation of analytic functions by the complex q-Bernstein polynomials on compact disks. Finally, we study the approximation properties of their iterates for any q > 0 and we prove that the complex qn-Bernstein polynomials with 0 < qn < 1 and qn → 1, preserve in the unit disk (beginning with an index) the starlikeness, convexity and spiral-likeness.

Approximation by complex Chlodowsky-Szasz-Durrmeyer operators in compact disks

2020 ◽  
Vol 29 (1) ◽  
pp. 37-44
Author(s):  
AYDIN IZGI ◽  
SEVILAY KIRCI SERENBAY
Keyword(s):  
Error Estimates ◽  
Exponential Growth ◽  
Analytic Functions ◽  
Approximation Properties ◽  
Type Result ◽  
Durrmeyer Operators ◽  
Numerical Example ◽  
Compact Disks

This paper presents a study on the approximation properties of the operators constructed by the composition of Chlodowsky operators and Szasz-Durrmeyer operators. We give the approximation properties and obtain a Voronovskaya-type result for these operators for analytic functions of exponential growth on compact disks. Furthermore, a numerical example with an illustrative graphic is given to compare for the error estimates of the operators.

Approximation by complex summation-integral type operator in compact disks

2013 ◽  
Vol 63 (5) ◽  
Author(s):  
Vijay Gupta ◽  
Rani Yadav
Keyword(s):  
Complex Plane ◽  
Analytic Functions ◽  
Quantitative Estimate ◽  
Type Operator ◽  
Approximation Properties ◽  
Integral Type ◽  
Quantitative Estimates ◽  
Compact Disks ◽  
Durrmeyer Polynomials

AbstractIn the present paper we estimate a Voronovskaja type quantitative estimate for a certain type of complex Durrmeyer polynomials, which is different from those studied previously in the literature. Such estimation is in terms of analytic functions in the compact disks. In this way, we present the evidence of overconvergence phenomenon for this type of Durrmeyer polynomials, namely the extensions of approximation properties (with quantitative estimates) from real intervals to compact disks in the complex plane. In the end, we mention certain applications.

Blending type approximation by complex Szász-Durrmeyer-Chlodowsky operators in compact disks

2019 ◽  
Vol 69 (5) ◽  
pp. 1077-1088
Author(s):  
Meenu Goyal ◽  
P. N. Agrawal
Keyword(s):  
Analytic Functions ◽  
Simultaneous Approximation ◽  
Asymptotic Result ◽  
Exact Order ◽  
Approximation Properties ◽  
Type Result ◽  
Type Approximation ◽  
Quantitative Estimates ◽  
Compact Disks ◽  
Upper Estimate

Abstract In the present article, we deal with the overconvergence of the Szász-Durrmeyer-Chlodowsky operators. Here we study the approximation properties e.g. upper estimates, Voronovskaja type result for these operators attached to analytic functions in compact disks. Also, we discuss the exact order in simultaneous approximation by these operators and its derivatives and the asymptotic result with quantitative upper estimate. In such a way, we put in evidence the overconvergence phenomenon for the Szász-Durrmeyer-Chlodowsky operators, namely the extensions of approximation properties with exact quantitative estimates and orders of these convergencies to sets in the complex plane that contain the interval [0, ∞).

scholarly journals Approximation by a power series summability method of Kantorovich type Szász operators including Sheffer polynomials

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Valdete Loku ◽  
Naim L. Braha ◽  
Toufik Mansour ◽  
M. Mursaleen
Keyword(s):  
Power Series ◽  
Summability Method ◽  
Approximation Properties ◽  
Type Result ◽  
Szász Operators ◽  
Sheffer Polynomials

AbstractThe main purpose of this paper is to use a power series summability method to study some approximation properties of Kantorovich type Szász–Mirakyan operators including Sheffer polynomials. We also establish Voronovskaya type result.

scholarly journals A stochastic proof of an extension of a theorem of Rado

1983 ◽  
Vol 26 (3) ◽  
pp. 333-336 ◽  
Author(s):  
Bernt Øksendal
Keyword(s):  
Brownian Motion ◽  
Analytic Function ◽  
Meromorphic Function ◽  
Conformal Invariance ◽  
Analytic Functions ◽  
Removable Singularity ◽  
Boundary Behaviour ◽  
Type Result ◽  
Cluster Set ◽  
Asymptotic Values

The purpose of this article is to illustrate how the theorem of Lévy about conformal invariance of Brownian motion can be used to obtain information about boundary behaviour and removable singularity sets of analytic functions. In particular, we prove a Frostman–Nevanlinna–Tsuji type result about the size of the set of asymptotic values of an analytic function at a subset of the boundary of its domain of definition (Theorem 1). Then this is used to prove the following extension of the classical Radó theorem: If φ is analytic in B\K, where B is the unit ball of ℂ;n and K is a relatively closed subset of B, and the cluster set of φ at K has zero harmonic measure w.r.t. φ(B\K)\≠Ø, then φ extends to a meromorphic function in B (Theorem 2).

scholarly journals Approximation of functions by a new class of linear operators

1972 ◽  
Vol 13 (3) ◽  
pp. 271-276 ◽  
Author(s):  
G. C. Jain
Keyword(s):  
Negative Binomial ◽  
Bernstein Polynomials ◽  
Degree Of Approximation ◽  
Linear Operators ◽  
Approximation Properties ◽  
Convergence Properties ◽  
Approximation Of Functions ◽  
New Class ◽  
Binomial Distributions ◽  
Poisson Type

Various extensions and generalizations of Bernstein polynomials have been considered among others by Szasz [13], Meyer-Konig and Zeller [8], Cheney and Sharma [1], Jakimovski and Leviatan [4], Stancu [12], Pethe and Jain [11]. Bernstein polynomials are based on binomial and negative binomial distributions. Szasz and Mirakyan [9] have defined another operator with the help of the Poisson distribution. The operator has approximation properties similar to those of Bernstein operators. Meir and Sharma [7] and Jam and Pethe [3] deal with generalizations of Szasz-Mirakyan operator. As another generalization, we define in this paper a new operator with the help of a Poisson type distribution, consider its convergence properties and give its degree of approximation. The results for the Szasz-Mirakyan operator can easily be obtained from our operator as a particular case.

scholarly journals On approximation properties of Baskakov-Schurer-Szász operators preserving exponential functions

Filomat ◽  
2018 ◽  
Vol 32 (15) ◽  
pp. 5433-5440 ◽  
Author(s):  
Övgü Yılmaz ◽  
Murat Bodur ◽  
Ali Aral
Keyword(s):  
Uniform Convergence ◽  
Generating Function ◽  
General Class ◽  
Quantitative Estimate ◽  
Moment Generating Function ◽  
Convergence Result ◽  
Weighted Spaces ◽  
Approximation Properties ◽  
Exponential Functions ◽  
Szász Operators

The goal of this paper is to construct a general class of operators which has known Baskakov-Schurer-Sz?sz that preserving constant and e2ax, a > 0 functions. Also, we demonstrate the fact that for these operators, moments can be obtained using the concept of moment generating function. Furthermore, we investigate a uniform convergence result and a quantitative estimate in consideration of given operator, as well. Finally, we discuss the convergence of corresponding sequences in exponential weighted spaces and make a comparison about which one approximates better between classical Baskakov-Schurer-Sz?sz operators and the recent sequence, too.

scholarly journals Yet Another New Variant of Szász–Mirakyan Operator

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2018
Author(s):  
Ana Maria Acu ◽  
Gancho Tachev
Keyword(s):  
Quantitative Estimate ◽  
Type Result ◽  
New Variant

In this paper, we construct a new variant of the classical Szász–Mirakyan operators, Mn, which fixes the functions 1 and eax,x≥0,a∈R. For these operators, we provide a quantitative Voronovskaya-type result. The uniform weighted convergence of Mn and a direct quantitative estimate are obtained. The symmetry of the properties of the classical Szász–Mirakyan operator and of the properties of the new sequence is investigated. Our results improve and extend similar ones on this topic, established in the last decade by many authors.

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