Probability Techniques in Analysis and Approximation Theory

Hyperbolic Fourier series and the Klein--Gordon equation

28 Nov 2024, 17:00

1h

Saint-Petersburg University

Speaker

Haakan Hedenmalm (Royal Institute of Technology (KTH), Stockholm, and Saint Petersburg State University)

Description

The Klein--Gordon equation in 1+1 dimensions is one of the truly basic second order PDEs with constant coefficients.
It models the time evolution of a one-dimensional relativistic boson with spin 0. Since it is relativisitic, the temporal relation between points is felt, and a given pair of points is either time-like or space-like. If the pair of points is space-like, we cannot say that one or the other event happens before or after the other.
If we study a space-like cone, and place equidistributed points on the edges, do we get a uniqueness set for Klein--Gordon solutions?
The answer turns out to depend on the density of points, and the shape of the solution.

As a consequence, we are led to study hyperbolic Fourier series, a topic which is natural but is a recent discovery only. The first installment is a paper with A. Montes-Rodriguez (Annals of Mathematics, 2011). The second only exists as
a 2024 preprint, but it builds on insights in the work of Radchenko and Viazovska (2019).