Gap and Type problems in Fourier analysis
Gap and type problems originated in the work of Norbert Wiener, Andrei Kolmogorov and Mark Krein in the 1930s and 40s. The Uncertainty Principle in harmonic analysis (UP), as formulated by Wiener around 1925, says that a function (measure, distribution) and its Fourier transform cannot be simultaneously small. The gap problem can be viewed as a statement of UP saying that a measure and its Fourier transform cannot both have small supports, in the sense that if the support of the measure is porous then its Fourier spectrum cannot have large gaps. The type problem, usually stated as a problem of completeness of trigonometric polynomials in L^2-spaces, becomes a version of the gap problem via duality. In my talk I will present the history of both problems, their connections to adjacent fields of analysis and recent solutions.