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Inclusion isotonicity of convex-concave extensions for polynomials based on Bernstein expansion
Publikationstyp
Journal Article
Date Issued
2003-04-07
Sprache
English
Institut
TORE-URI
Journal
Volume
70
Issue
2
Start Page
111
End Page
119
Citation
Computing (Vienna/New York) 2 (70): 111-119 (2003-08-07)
Publisher DOI
Scopus ID
Publisher
Springer
In this paper the expansion of a polynomial into Bernstein polynomials over an interval I is considered. The convex hull of the control points associated with the coefficients of this expansion encloses the graph of the polynomial over I. By a simple proof it is shown that this convex hull is inclusion isotonic, i.e. if one shrinks I then the convex hull of the control points on the smaller interval is contained in the convex hull of the control points on I. From this property it follows that the so-called Bernstein form is inclusion isotone, which was shown by a longish proof in 1995 in this journal by Hong and Stahl. Inclusion isotonicity also holds for multivariate polynomials on boxes. Examples are presented which document that two simpler enclosures based on only a few control points are in general not inclusion isotonic.
Subjects
Bernstein polynomials
Bound functions
Control points
Convex hull
Inclusion isotonicity
DDC Class
004: Informatik
510: Mathematik
More Funding Information
The authors gratefully acknowledge support from the Ministry of Education and Research of the Federal Republic of Germany under contract no. 1707001.