The Fundamental Blossoming Inequality in Chebyshev Spaces—I: Applications to Schur Functions

Handle URI:
http://hdl.handle.net/10754/622257
Title:
The Fundamental Blossoming Inequality in Chebyshev Spaces—I: Applications to Schur Functions
Authors:
Ait-Haddou, Rachid; Mazure, Marie Laurence
Abstract:
A classical theorem by Chebyshev says how to obtain the minimum and maximum values of a symmetric multiaffine function of n variables with a prescribed sum. We show that, given two functions in an Extended Chebyshev space good for design, a similar result can be stated for the minimum and maximum values of the blossom of the first function with a prescribed value for the blossom of the second one. We give a simple geometric condition on the control polygon of the planar parametric curve defined by the pair of functions ensuring the uniqueness of the solution to the corresponding optimization problem. This provides us with a fundamental blossoming inequality associated with each Extended Chebyshev space good for design. This inequality proves to be a very powerful tool to derive many classical or new interesting inequalities. For instance, applied to Müntz spaces and to rational Müntz spaces, it provides us with new inequalities involving Schur functions which generalize the classical MacLaurin’s and Newton’s inequalities. This work definitely demonstrates that, via blossoms, CAGD techniques can have important implications in other mathematical domains, e.g., combinatorics.
KAUST Department:
Visual Computing Center (VCC); Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Citation:
Ait-Haddou R, Mazure M-L (2016) The Fundamental Blossoming Inequality in Chebyshev Spaces—I: Applications to Schur Functions. Foundations of Computational Mathematics. Available: http://dx.doi.org/10.1007/s10208-016-9334-8.
Publisher:
Springer Nature
Journal:
Foundations of Computational Mathematics
Issue Date:
19-Oct-2016
DOI:
10.1007/s10208-016-9334-8
Type:
Article
ISSN:
1615-3375; 1615-3383
Appears in Collections:
Articles; Visual Computing Center (VCC); Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorAit-Haddou, Rachiden
dc.contributor.authorMazure, Marie Laurenceen
dc.date.accessioned2017-01-02T08:42:40Z-
dc.date.available2017-01-02T08:42:40Z-
dc.date.issued2016-10-19en
dc.identifier.citationAit-Haddou R, Mazure M-L (2016) The Fundamental Blossoming Inequality in Chebyshev Spaces—I: Applications to Schur Functions. Foundations of Computational Mathematics. Available: http://dx.doi.org/10.1007/s10208-016-9334-8.en
dc.identifier.issn1615-3375en
dc.identifier.issn1615-3383en
dc.identifier.doi10.1007/s10208-016-9334-8en
dc.identifier.urihttp://hdl.handle.net/10754/622257-
dc.description.abstractA classical theorem by Chebyshev says how to obtain the minimum and maximum values of a symmetric multiaffine function of n variables with a prescribed sum. We show that, given two functions in an Extended Chebyshev space good for design, a similar result can be stated for the minimum and maximum values of the blossom of the first function with a prescribed value for the blossom of the second one. We give a simple geometric condition on the control polygon of the planar parametric curve defined by the pair of functions ensuring the uniqueness of the solution to the corresponding optimization problem. This provides us with a fundamental blossoming inequality associated with each Extended Chebyshev space good for design. This inequality proves to be a very powerful tool to derive many classical or new interesting inequalities. For instance, applied to Müntz spaces and to rational Müntz spaces, it provides us with new inequalities involving Schur functions which generalize the classical MacLaurin’s and Newton’s inequalities. This work definitely demonstrates that, via blossoms, CAGD techniques can have important implications in other mathematical domains, e.g., combinatorics.en
dc.publisherSpringer Natureen
dc.subjectChebyshevian blossomingen
dc.subjectExtended Chebyshev spacesen
dc.subjectMacLaurin’s inequalitiesen
dc.subjectMüntz spacesen
dc.subjectNewton’s inequalitiesen
dc.subjectNormalized Schur functionsen
dc.titleThe Fundamental Blossoming Inequality in Chebyshev Spaces—I: Applications to Schur Functionsen
dc.typeArticleen
dc.contributor.departmentVisual Computing Center (VCC)en
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.identifier.journalFoundations of Computational Mathematicsen
dc.contributor.institutionLaboratoire Jean Kuntzmann, CNRS, UMR 5224, Université Grenoble Alpes, Campus de Saint Martin d’Hères, Grenoble, 38041, Franceen
kaust.authorAit-Haddou, Rachiden
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