Approximation properties of rth order generalized Bernstein polynomials based on q-calculus

This paper is concerned with a generalization of the classical Bernstein polynomials where the function is evaluated at intervals which are in geometric progression. It is shown that, when the function is convex, the generalized Bernstein polynomials Bn are monotonic in n, as in the classical case.

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

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.2010.1156 ◽

2011 ◽

Vol 48 (1) ◽

pp. 23-43 ◽

Cited By ~ 3

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.

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Convergence of Generalized Bernstein Polynomials

Journal of Approximation Theory ◽

10.1006/jath.2001.3657 ◽

2002 ◽

Vol 116 (1) ◽

pp. 100-112 ◽

Cited By ~ 93

Author(s):

Alexander Il'inskii ◽

Sofiya Ostrovska

Keyword(s):

Bernstein Polynomials ◽

Generalized Bernstein Polynomials

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Approximation of functions by a new class of linear operators

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700013689 ◽

1972 ◽

Vol 13 (3) ◽

pp. 271-276 ◽

Cited By ~ 48

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.

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