The exact order of approximation of functions by Bernstein polynomials in a Hausdorff metric

1972 ◽  
Vol 12 (5) ◽  
pp. 737-742 ◽  
Author(s):  
V. M. Veselinov
Keyword(s):  
Bernstein Polynomials ◽  
Hausdorff Metric ◽  
Exact Order ◽  
Order Of Approximation ◽  
Approximation Of Functions ◽  
Exact Order Of Approximation

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 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.

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.

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 Approximation characteristics of the isotropic Nikol'skii-Besov functional classes

2021 ◽  
Vol 13 (3) ◽  
pp. 851-861
Author(s):  
S.Ya. Yanchenko ◽  
O.Ya. Radchenko
Keyword(s):  
Exponential Type ◽  
Lebesgue Space ◽  
Entire Functions ◽  
Exact Order ◽  
Periodic Functions ◽  
Approximation Of Functions ◽  
Several Variables ◽  
Functions Of Exponential Type ◽  
Functional Classes ◽  
Mixed Smoothness

In the paper, we investigates the isotropic Nikol'skii-Besov classes $B^r_{p,\theta}(\mathbb{R}^d)$ of non-periodic functions of several variables, which for $d = 1$ are identical to the classes of functions with a dominant mixed smoothness $S^{r}_{p,\theta}B(\mathbb{R})$. We establish the exact-order estimates for the approximation of functions from these classes $B^r_{p,\theta}(\mathbb{R}^d)$ in the metric of the Lebesgue space $L_q(\mathbb{R}^d)$, by entire functions of exponential type with some restrictions for their spectrum in the case $1 \leqslant p \leqslant q \leqslant \infty$, $(p,q)\neq \{(1,1), (\infty, \infty)\}$, $d\geq 1$. In the case $2<p=q<\infty$, $d=1$, the established estimate is also new for the classes $S^{r}_{p,\theta}B(\mathbb{R})$.

scholarly journals Automatic choice of parameters at approximation of functions by rational splines

1987 ◽  
pp. 52
Author(s):  
A.D. Malysheva
Keyword(s):  
Sufficient Conditions ◽  
Order Of Approximation ◽  
Approximation Of Functions ◽  
Smooth Functions ◽  
Higher Derivatives ◽  
Necessary And Sufficient ◽  
Automatic Choice ◽  
Derivatives Of ◽  
Best Parameters ◽  
Approximation Of A Function

We obtain necessary and sufficient conditions put on the parameters of rational splines that provide given order of approximation of smooth functions. We point out the formulas of asymptotically the best parameters of rational splines that, while providing the best order of approximation of a function by rational splines, do not contain information about the values of higher derivatives of a function.

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