### Extension of Bernstein's theorem to Sturm-Liouville sums

Elizabeth Carlson
1924 Transactions of the American Mathematical Society
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.
more » ... o great as 2nL. Various proofs were given by later writers,^ leading to the simplified statement which appears above. The simplest proof was discovered independently by Marcel Riess § and de la Vallée Poussin.|| The purpose of this paper is to prove the corresponding theorem for Sturm-Liouville sums: The maximum of the absolute value of the derivative of a Sturm-Liouville sum of order n(w ^> 1) can not exceed npM, where M is the maximum of the absolute value of the sum itself, and p is independent of n and of the coefficients in the sum.