Chebyshev and Equilibrium Measure Vs Bernstein and Lebesgue Measure
Abstract
We show that Bernstein polynomials are related to the Lebesgue measure on [0, 1] in a manner similar as Chebyshev polynomials are related to the equilibrium measure of [--1, 1]. We also show that Pell's polynomial equation satisfied by Chebyshev polynomials, provides a partition of unity of [--1, 1], the analogue of the partition of unity of [0, 1] provided by Bernstein polynomials. Both partitions of unity are interpreted as a specific algebraic certificate that the constant polynomial ''1'' is positive-on [--1, 1] via Putinar's certificate of positivity (for Chebyshev), and-on [0, 1] via Handeman's certificate of positivity (for Bernstein). Then in a second step, one combines this partition of unity with an interpretation of a duality result of Nesterov in convex conic optimization to obtain an explicit connection with the equilibrium measure on [--1, 1] (for Chebyshev) and Lebesgue measure on [0, 1] (for Bernstein). Finally this connection is also partially established for the ''d''-dimensional simplex.
Keywords
Cite
@article{arxiv.2303.13975,
title = {Chebyshev and Equilibrium Measure Vs Bernstein and Lebesgue Measure},
author = {Jean-Bernard Lasserre},
journal= {arXiv preprint arXiv:2303.13975},
year = {2023}
}