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- W4360978312 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." @default.
- W4360978312 created "2023-03-30" @default.
- W4360978312 creator A5006670993 @default.
- W4360978312 date "2023-03-01" @default.
- W4360978312 modified "2023-10-16" @default.
- W4360978312 title "CHEBYSHEV AND EQUILIBRIUM MEASURE VS BERNSTEIN AND LEBESGUE MEASURE" @default.
- W4360978312 hasPublicationYear "2023" @default.
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