On the Convergence and Iterates of q-Bernstein Polynomials

Oruç, Halil; Tuncer, Necibe

doi:10.1006/jath.2002.3703

Cited by 42 publications

(31 citation statements)

References 13 publications

(11 reference statements)

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“…Necessary and sufficient conditions for the convergence of (P n ) n∈N are well known (see, e.g., [Ga59, Chapter 13]) and can be applied here. Actually, such an approach was also used in [KR67,OT02,We97], but in those papers it led to arduous calculations. Our method is more abstract and it seems to be simpler.…”

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confidence: 99%

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Convergence of iterates of linear operators and the Kelisky–Rivlin type theorems

Jachymski

2009

Studia Math.

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Abstract. Let X be a Banach space and T ∈ L(X), the space of all bounded linear operators on X. We give a list of necessary and sufficient conditions for the uniform stability of T , that is, for the convergence of the sequence (T n ) n∈N of iterates of T in the uniform topology of L(X). In particular, T is uniformly stable iff for some p ∈ N, the restriction of the pth iterate of T to the range of I − T is a Banach contraction. Our proof is elementary: It uses simple facts from linear algebra, and the Banach Contraction Principle. As a consequence, we obtain a theorem on the uniform convergence of iterates of some positive linear operators on C(Ω), which generalizes and subsumes many earlier results including, the Kelisky-Rivlin theorem for univariate Bernstein operators, and its extensions for multivariate Bernstein polynomials over simplices. As another application, we also get a new theorem in this setting giving a formula for the limit of iterates of the tensor product Bernstein operators.

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confidence: 99%

“…3]). There are a number of results in the literature (see, e.g., [AA96], [AR03], [CF86], [CF93], [GP05], [KR67], [OT02], [Ru04], [We97]) showing that some particular linear operators are strongly stable, and giving a formula for the operator T ∞ . For example, the following theorem was proved in [KR67].…”

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confidence: 99%

Convergence of iterates of linear operators and the Kelisky–Rivlin type theorems

Jachymski

2009

Studia Math.

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“…Oruc, Tuncer in [3] have presented some results about the convergence of B M n ( f , q; x) and M B n ( f , q; x) as M → ∞ as follows.…”

Section: The Boolean Sum Of Two Operators a And B Is Defined Bymentioning

confidence: 97%

“…Yet, in [10] Ostrovska has studied the analytic properties of limit q-Bernstein operators. Lately, Oruc and Tuncer in [3] have discussed convergence properties for iterates of both q-Bernstein polynomials and their Boolean sum. The convergence of B n ( f , q; x) as q → ∞ along with their iterates B j n n ( f , q; x) as both n → ∞ and j n → ∞, have been investigated in [4] by Ostrovska. In this paper, we give the saturation of {B n (·, q n )} when n → ∞ and the convergence rate of B n ( f , q; x) for f ∈ C n−1 [0, 1], when q → ∞.…”

Section: Introductionmentioning

confidence: 99%

Convergence rate for iterates of q-Bernstein polynomials

Xiang

Yang

2007

Analys in Theo Applic

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Recently, q-Bernstein polynomials have been intensively investigated by a number of authors. Their results show that for q = 1, q-Bernstein polynomials possess of many interesting properties. In this paper, the convergence rate for iterates of both q-Bernstein polynomials and their Boolean sum are estimated. Moreover, the saturation of {B n (·, q n )} when n → ∞ and convergence rate of B n ( f , q; x) when f ∈ C n−1 [0, 1], q → ∞ are also presented.

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“…Research results show that q-Bernstein operators possess good convergence and approximation properties in C [0, 1]. These operators have been studied by a number of authors, we mention the some due to II'inskii and Ostrovska [16], Oruc and Tuncer [21], Ostrovska [22], [23] and Videnskii [28] etc. Heping [14], Heping and Fanjun [15] discussed Voronovskaya-type formulas and saturation of convergence for q-Bernstein polynomials for arbitrary fixed q, 0 < q < 1.…”

Section: Introductionmentioning

confidence: 99%