TY - JOUR

T1 - The Fundamental Blossoming Inequality in Chebyshev Spaces—I: Applications to Schur Functions

AU - Ait-Haddou, Rachid

AU - Mazure, Marie Laurence

N1 - KAUST Repository Item: Exported on 2020-10-01

PY - 2016/10/19

Y1 - 2016/10/19

N2 - 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.

AB - 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.

UR - http://hdl.handle.net/10754/622257

UR - http://link.springer.com/10.1007/s10208-016-9334-8

UR - http://www.scopus.com/inward/record.url?scp=84991798501&partnerID=8YFLogxK

U2 - 10.1007/s10208-016-9334-8

DO - 10.1007/s10208-016-9334-8

M3 - Article

SN - 1615-3375

VL - 18

SP - 135

EP - 158

JO - Foundations of Computational Mathematics

JF - Foundations of Computational Mathematics

IS - 1

ER -