Paper Info
Title
The Fundamental Blossoming Inequality in Chebyshev Spaces - I: Applications to Schur Functions.
Abstract
A classical theorem by Chebyshev says how to obtain the minimum and maximum values of a symmetric multiaffine function of 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.
Year
DOI
Venue
2018
https://doi.org/10.1007/s10208-016-9334-8
Foundations of Computational Mathematics
Keywords
Field
DocType
Extended Chebyshev spaces,Chebyshevian blossoming,MacLaurin’s inequalities,Newton’s inequalities,Müntz spaces,Normalized Schur functions,41A05,41A10,41A29,65D05,65D17,65D18
Uniqueness,Polygon,Mathematical optimization,Parametric equation,Mathematical analysis,Chebyshev's sum inequality,Markov's inequality,Chebyshev filter,Multidimensional Chebyshev's inequality,Optimization problem,Mathematics
Journal
Volume
Issue
ISSN
18
1
1615-3375
Citations 
PageRank 
References 
0
0.34
10
Authors
2
Name
Order
Citations
PageRank
Rachid Ait-Haddou141.15
Marie-Laurence Mazure223428.01