Cours fondamental 2 (HFE)

Uncertainty principles and their relations to Remez-type inequalities

Evgeny Abakumov et Omer Friedland

Contact : omer.friedland à imj-prg.fr

Pas de notes de cours prévues.

Présentation

The uncertainty principle in Harmonic Analysis is a collection of related results which give sense to the following proposition: one cannot concentrate the mass of a function and its Fourier transform simultaneously. In these lectures, we'll concentrate on one of its numerous manifestations, namely, Logvinenko-Sereda Theorem, its generalizations and the application to local solvability of constant-coefficient linear PDE.

In the first part of the lectures we consider some classical polynomial inequalities, e.g. Remez inequality. We plan to discuss some equivalent statements of this result and present some classical generalizations, like the Tur\'an-Nazarov inequality. We shall extend these results and show that the Lebesgue measure in these inequalities can be replaced with a certain geometric invariant, which can be effectively estimated in terms of the metric entropy of a set, and may be non-zero for discrete sets and even finite sets.

In the second part, we provide some applications which are related to the Logvinenko-Sereda theorem which give limitations on the simultaneous concentration of a function and its Fourier transform. We present certain generalizations of this theorem for different classes of functions which satisfy a Remez-type inequality, and discuss applications to non-controllability of some PDE.

The course is useful for doctoral students and advanced master's students in mathematics, and other related fields, who are looking to broaden their knowledge in modern analysis in light of classical results and methods.