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One of the most important of recent theorems in analysis is a theorem due to S. Bernstein, which may be stated as follows: If Tn(x) is a trigonometric sum of order n, the maximum of whose absolute value does not exceed L, then the maximum of the absolute value of the derivative Tn(x) does not exceed nL. Bernsteinf proved the corresponding theorem for polynomials first, and from it obtained the theorem for the trigonometric case. His conclusion was that |Tn(a;)| could not be so great as 2nL.doi:10.1090/s0002-9947-1924-1501275-5 fatcat:w3xhxvaxrrbnngkdd5p33xmhoe