On the number of sign changes of a real function

Abstract

The aim of this note is to give a lower bound for the number of sign changes of a real function. This bound depends on the sequence of moments. Several results are known along these lines. However, to the knowledge of the author, all of them focus on real functions with nonnegative support (Fej~r [2],P61ya--Szeg6 [6], B6dewadt [1], Obreschkoff [5]). Some of the methods that have been used, e.g. Laplace transforms, do not apply to functions with support [a,b], ~ < a < b < ~ , if a ~ 0 ~ b . I t is obvious that in this setting, the results obtainable are weaker than those mentioned above. A vanishing first moment does not any more imply a sign change of the function in contrast to the situation where we consider a function with nonnegative support. The following result relates rather general structures in the sequence of moments to the number of sign changes. Some special cases are discussed. The proof is based on elementary techniques.