The two sides of a fourier-stieltjes transform and almost idempotent measures

Abstract

For measuresμ on the circleT the quantities$lim sup_{n \to + \infty } \left| {\hat \mu (n)} \right|$,$lim sup\left| {\hat \mu (n)} \right|_{_{n \to - \infty } }$ need not be equal; it is shown, however, that they are continuous with respect to each other whenμ varies on bounded subsets ofM(T), the space of measures onT. It is also shown that measuresμ which areɛ-almost idempotent (i.e.$lim sup_{\left| n \right| \to \infty } \left| {\hat \mu (n) - \hat \mu (n)^2 } \right|< \varepsilon$) are the sum of an idempotent measure and of a measureυ satisfying$lim sup_{\left| n \right| \to \infty } \left| {\hat \nu (n)} \right|< 2\varepsilon$ providedɛ is small enough (as a function of ‖μ‖).