Chebyshev and Equilibrium Measure Vs Bernstein and Lebesgue Measure
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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.
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