Sign uncertainty principles for entire functions of exponential type

The Fourier uncertainty principle is a concept that arises in various forms. In a nutshell, it describes the inability to control a function and its Fourier transform at the same time. A significant variant, introduced by Bourgain, Clozel, and Kahane [1], is the sign uncertainty principle. In simple terms, their result states that if both a function and its Fourier transform are eventually non-negative, their negative masses cannot be “too concentrated” near the origin simultaneously. In this talk, we will address sign uncertainty principles for eventually non-negative functions of exponential type under various measure constraints. I will present how these uncertainty principles fit in the framework of Hilbert spaces of entire functions and some of their applications to number theory.
This talk is based on joint work with E.Carneiro and A.P.Ramos [2].

References

• J.Bourgain, L.Clozel, and J.-P.Kahane. Principe d’Heisenberg et fonctions positives, Annales de l’institut Fourier, 60(4):1215–1232, 2010.
• E.Carneiro, T.Ismoilov, and A.P.Ramos. Sign uncertainty and de Branges spaces, PrePrint, 2024.