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

Type
Article

Authors
Ait-Haddou, Rachid
Mazure, Marie Laurence

KAUST Department
Visual Computing Center (VCC)
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Online Publication Date
2016-10-19

Print Publication Date
2018-02

Date
2016-10-19

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.

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

DOI
10.1007/s10208-016-9334-8

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