Extension of Bernstein’s theorem to Sturm-Liouville sums

Abstract

One of the most important of recent theorems in analysis is a theorem due to S. Bernstein, which may be stated as follows: If T, (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. Bernsteint proved the corresponding theorem for polynomials first, and from it obtained the theorem for the trigonometric case. His conclusion was that IT.(x)l could not be so 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 Vallee Poussin.11 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 sutm of order n(n ? 1) can not exceed np M, wvhere M is the maximum of the absolute value of the sium itself, and p is independent of n and of the coefficients in the sum. The proof to be given here is similar to one which de la Vallee Poussin?T