Shape preserving properties of generalized Bernstein operators on extended Chebyshev spaces

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Title: Shape preserving properties of generalized Bernstein operators on extended Chebyshev spaces
Authors: Aldaz, J. M.
Kounchev, Ognyan
Render, Hermann
Permanent link: http://hdl.handle.net/10197/5495
Date: Dec-2009
Online since: 2014-03-28T09:05:29Z
Abstract: We study the existence and shape preserving properties of a generalized Bernstein operator B n fixing a strictly positive function f 0 , and a second function f 1 such that f 1 /f 0 is strictly increasing, within the framework of extended Chebyshev spaces U n . The first main result gives an inductive criterion for existence: suppose there exists a Bernstein operator B n : C [ a,b ] → U n with strictly increasing nodes, fixing f 0 ,f 1 ∈ U n . If U n ⊂ U n +1 and U n +1 has a non-negative Bernstein basis, then there exists a Bernstein operator B n +1 : C [ a,b ] → U n +1 with strictly increasing nodes, fixing f 0 and f 1 . In particular, if f 0 ,f 1 ,...,f n is a basis of U n such that the linear span of f 0 ,..,f k is an extended Chebyshev space over [ a,b ] for each k = 0 ,...,n , then there exists a Bernstein operator B n with increasing nodes fixing f 0 and f 1 . The second main result says that under the above assumptions the following inequalities hold B n f ≥ B n +1 f ≥ f for all ( f 0 ,f 1 )-convex functions f ∈ C [ a,b ] . Furthermore, B n f is ( f 0 ,f 1 )-convex for all ( f 0 ,f 1 )-convex functions f ∈ C [ a,b ] .
Type of material: Journal Article
Publisher: Springer
Journal: Numerische Mathematik
Volume: 114
Issue: 1
Start page: 1
End page: 25
Copyright (published version): 2009 Springer
Keywords: Bernstein operatorsChebyshev spaces
DOI: 10.1007/s00211-009-0248-0
Language: en
Status of Item: Peer reviewed
Appears in Collections:Mathematics and Statistics Research Collection

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