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.

翻译：我们证明：伯恩斯坦多项式与[0,1]上的勒贝格测度的关系，类似于切比雪夫多项式与[-1,1]上平衡测度的关系。同时表明，切比雪夫多项式所满足的佩尔多项式方程给出了[-1,1]上的单位分解，这与伯恩斯坦多项式提供的[0,1]上的单位分解类似。两种单位分解均被解释为具体的代数证明：常数多项式“1”在[-1,1]上（通过普蒂纳尔正性证明，对应切比雪夫）和在[0,1]上（通过汉德曼正性证明，对应伯恩斯坦）具有正性。进而，我们结合该单位分解与涅斯捷罗夫在凸锥优化中对偶性结果的一个解释，得到切比雪夫情形下[-1,1]上平衡测度与伯恩斯坦情形下[0,1]上勒贝格测度的显式联系。最后，该联系在d维单纯形上也得到了部分建立。