Lower bounds for quasianalytic functions, II The Bernstein quasianalytic functions.

Abstract

Let $\mathscr{F}$ be a class of functions with the uniqueness property: if $f \in \mathscr{F}$ vanishes on a set $E$ of positive measure, then $f$ is the zero function. In many instances, we would like to have a quantitative version of this property, e.g. a lower bound for $|f|$ outside a small exceptional set. Such estimates are well-known and useful for polynomials, complex- and real-analytic functions, exponential polynomials. In this work we prove similar results for the Denjoy-Carleman and the Bernstein classes of quasianalytic functions. In the first part, we considered quasianalytically smooth functions. Here, we deal with classes of functions characterized by exponentially fast approximation by polynomials whose degrees belong to a given very lacunar sequence. We also prove the polynomial spreading lemma and a comparison lemma which are of a certain interest on their own.