Article Dans Une Revue Proceedings of the American Mathematical Society Année : 2024

CHEBYSHEV AND EQUILIBRIUM MEASURE VS BERNSTEIN AND LEBESGUE MEASURE

Résumé

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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Dates et versions

hal-04043186 , version 1 (23-03-2023)

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Jean-Bernard Lasserre. CHEBYSHEV AND EQUILIBRIUM MEASURE VS BERNSTEIN AND LEBESGUE MEASURE. Proceedings of the American Mathematical Society, In press, ⟨10.1090/proc/16739⟩. ⟨hal-04043186⟩
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