Approximation by Complex Bernstein and Convolution Type Operators

Gal, Sorin G.

doi:10.1142/7426

Cited by 99 publications

(77 citation statements)

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“…It is worth mentioning that in the univariate case the results similar to Theorem 1 were obtained by Ostrovska [5] and Gal [2]. On the other hand in the classical case (q = 1) the similar result is given in [2].…”

Section: Introduction and Main Resultssupporting

confidence: 71%

“…and attached to an analytic function f in a closed disk, were intensively studied by many authors, see [2] and the references theirin. It is known that the cases 0 < q < 1 and q > 1 are not similar to each other.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Our result is new even for the univariate case since the estimate given in Theorem 3 has quantitative character. The classical case was studied by Gal [2].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Approximation properties of bivariate complex q-Bernstein polynomials in the case q > 1

Mahmudov

2012

Czech Math J

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Section: Introduction and Main Resultssupporting

confidence: 71%

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Approximation properties of bivariate complex q-Bernstein polynomials in the case q > 1

Mahmudov

2012

Czech Math J

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“…The approximation by the q-Bernstein polynomials of quaternion variable corresponding to the case q = 1 in Definition 1.3, was studied in Section 4.1 of the book Gal [1], where the much more weaker approximation order 1 n was obtained.…”

Section: Sg Galmentioning

confidence: 99%

Approximation by Quaternion q-Bernstein Polynomials, q > 1

Gal

2011

Adv. Appl. Clifford Algebras

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Defining for q > 1 the q-Bernstein polynomials of degree n of a quaternion variable, attached to a function f defined on a ball in the field of quaternions, the order of approximation 1 q n is obtained when f is in some classes of analytic functions in the sense of Weierstrass. The result extends that in the case of approximation of analytic functions of a complex variable in disks, by q-Bernstein polynomials of complex variable.Keywords. Quaternion, q-Bernstein-polynomials of quaternion variable, order of approximation, analytic functions in Weierstrass sense.

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“…C is analytic in D R and all its derivatives bounded on [0; 1) by the same constant and also has an expontential growth condition for all z 2 D R and for j = 0; one takes n(n + 1):::(n + j 1) = 1. In [8], the Voronovskaja type results with a quantitative estimate. The exact order of simultaneous approximation for these operators were given.…”

Section: Introductionmentioning

confidence: 99%