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 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 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.

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

Filomat ◽  
2015 ◽  
Vol 29 (5) ◽  
pp. 1007-1019 ◽  
Author(s):  
Nursel Çetin
Keyword(s):  
Rate Of Convergence ◽  
Exponential Growth ◽  
Analytic Functions ◽  
Exact Order ◽  
Order Of Approximation ◽  
Stancu Operators ◽  
Exact Order Of Approximation ◽  
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.

scholarly journals Shape Preserving Properties forq-Bernstein-Stancu Operators

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Yali Wang ◽  
Yinying Zhou
Keyword(s):  
Bernstein Polynomials ◽  
Shape Preserving ◽  
Positive Basis ◽  
Stancu Operators ◽  
Variation Diminishing ◽  
Totally Positive ◽  
Shape Preserving Properties ◽  
Monotonicity Preserving

We investigate shape preserving forq-Bernstein-Stancu polynomialsBnq,α(f;x)introduced by Nowak in 2009. Whenα=0,Bnq,α(f;x)reduces to the well-knownq-Bernstein polynomials introduced by Phillips in 1997; whenq=1,Bnq,α(f;x)reduces to Bernstein-Stancu polynomials introduced by Stancu in 1968; whenq=1,α=0, we obtain classical Bernstein polynomials. We prove that basicBnq,α(f;x)basis is a normalized totally positive basis on[0,1]andq-Bernstein-Stancu operators are variation-diminishing, monotonicity preserving and convexity preserving on[0,1].

scholarly journals On Some Approximation Properties for a Sequence of λ-Bernstein Type Operators

2021 ◽  
pp. 4903-4915
Author(s):  
Ali Jassim Muhammad ◽  
Asma Jaber
Keyword(s):  
Asymptotic Formula ◽  
Uniform Convergence ◽  
Convergence Theorem ◽  
Recurrence Relations ◽  
Bernstein Polynomials ◽  
Negative Integer ◽  
Asymptotic Formulas ◽  
Order Moment ◽  
Approximation Properties ◽  
Bernstein Type

In 2010, Long and Zeng introduced a new generalization of the Bernstein polynomials that depends on a parameter  and called -Bernstein polynomials. After that, in 2018, Lain and Zhou studied the uniform convergence for these -polynomials and obtained a Voronovaskaja-type asymptotic formula in ordinary approximation. This paper studies the convergence theorem and gives two Voronovaskaja-type asymptotic formulas of the sequence of -Bernstein polynomials in both ordinary and simultaneous approximations. For this purpose, we discuss the possibility of finding the recurrence relations of the -th order moment for these polynomials and evaluate the values of -Bernstein for the functions ,  is a non-negative integer

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.

scholarly journals Convolution operators with trigonometric spline kernels

1988 ◽  
Vol 31 (2) ◽  
pp. 285-299 ◽  
Author(s):  
T. N. T. Goodman ◽  
S. L. Lee
Keyword(s):  
Algebraic Polynomial ◽  
Bernstein Polynomials ◽  
Spline Approximation ◽  
Convolution Operators ◽  
Shape Preserving ◽  
Approximation Operators ◽  
Trigonometric Spline ◽  
Polynomial Operators ◽  
Shape Preserving Properties ◽  
Schoenberg Spline

The Bernstein polynomials are algebraic polynomial approximation operators which possess shape preserving properties. These polynomial operators have been extended to spline approximation operators, the Bernstein-Schoenberg spline approximation operators, which are also shape preserving like the Bernstein polynomials [8].

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

Filomat ◽  
2015 ◽  
Vol 29 (5) ◽  
pp. 1127-1136
Author(s):  
Sorin Gal ◽  
Vijay Gupta
Keyword(s):  
Exponential Growth ◽  
Analytic Functions ◽  
Exact Order ◽  
Order Of Approximation ◽  
Durrmeyer Operators ◽  
Exact Order Of Approximation ◽  
Quantitative Estimates ◽  
Compact Disks

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.

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