Pólya peaks and the oscillation of positive functions

Abstract

A new proof is given of the existence of Pólya peaks of an (increasing) function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g left-parenthesis t right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">g(t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This approach yields several applications, including a characterization of those <italic>p</italic> for which <italic>g</italic> can have peaks of order <italic>p</italic>.