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

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

2003-01-20

arxiv.org/abs/math/0301217v2

Math. Scand. 95 (2004), 44--58.

Mathematics

Classical Analysis and ODEs

Scientific paper

Let F be a class of functions with the uniqueness property: if a function f in F vanishes on a set 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 (this arXiv:math.CA/0208233), 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.

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