Iterates of Bernstein polynomials

Kelisky, Richard P.; Rivlin, T. J.

doi:10.2140/pjm.1967.21.511

Cited by 112 publications

(53 citation statements)

References 1 publication

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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].…”

mentioning

confidence: 99%

See 1 more Smart Citation

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

Jachymski

2009

Studia Math.

View full text Add to dashboard Cite

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.

show abstract

mentioning

confidence: 99%

“…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.…”

mentioning

confidence: 99%

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

Jachymski

2009

Studia Math.

View full text Add to dashboard Cite

show abstract

“…In [11], Kelisky and Rivlin investigated the behavior of the iterates of the Bernstein polynomial of degree n ≥ 1 defined by …”

Section: Application: a Generalized Bernstein Operatormentioning

confidence: 99%

“…In particular, they used their ideas to discuss the iterate of the Bernstein operator. Moreover, they gave an example of a nonlinear version of the Bernstein operator and establish the Kelisky and Rivlin's theorem [11] for such operator.…”

Section: Introductionmentioning

confidence: 97%

A graphical version of Reichs fixed point theorem

Alfuraidan¹,

Bachar²,

Khamsi³

2016

J. Nonlinear Sci. Appl.

View full text Add to dashboard Cite

In this paper, we discuss the definition of the Reich multivalued monotone contraction mappings defined in a metric space endowed with a graph. In our investigation, we prove the existence of fixed point results for these mappings. We also introduce a vector valued Bernstein operator on the space C([0, 1], X), where X is a Banach space endowed with a partial order. Then we give an analogue to the Kelisky-Rivlin theorem.

show abstract

“…In recent years, q-Bernstein polynomials have been studied intensively by a number of authors. They investigated iterates properties of the Bernstein operator from a different point of view [4]- [6].…”

Section: Yali Wang and Yinying Zhoumentioning

confidence: 99%