Shape preserving properties of generalized Bernstein operators on extended Chebyshev spaces

Files in This Item:
File Description SizeFormat 
NMaAKRRev010109.pdf253.22 kBAdobe PDFDownload
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
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
Copyright (published version): 2009 Springer
Keywords: Bernstein operators;Chebyshev spaces
DOI: 10.1007/s00211-009-0248-0
Language: en
Status of Item: Peer reviewed
Appears in Collections:Mathematics and Statistics Research Collection

Show full item record

SCOPUSTM   
Citations 10

27
Last Week
0
Last month
checked on Jun 23, 2018

Page view(s) 50

35
checked on May 25, 2018

Download(s) 50

83
checked on May 25, 2018

Google ScholarTM

Check

Altmetric


This item is available under the Attribution-NonCommercial-NoDerivs 3.0 Ireland. No item may be reproduced for commercial purposes. For other possible restrictions on use please refer to the publisher's URL where this is made available, or to notes contained in the item itself. Other terms may apply.