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 $dx/(\pi \sqrt {1-x^2})$ of $[-1,1]$. We also show that polynomial Pell’s 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 simplex in $\mathbb {R}^d$.