BR Means

How do we truncate the inverse Fourier transform? This is a necessary question to ask as the inverse Fourier transform is expressed as an infinite integral. In practice we need to apply a truncation scheme to be able to effectively make computations. But we need to know whether our truncation scheme is convergent.

Fefferman famously proved that the ball truncation was not convergent in any L^p space except when p=2. Bochner-Riesz means where then introduced as a way to smooth out the ball truncation. One then asks, just how smooth do you need to be?

​In this project we are concerned with translating results about Bochner-Riesz means on Euclidean space into results on manifolds. On manifolds one of the key points to understand is the behaviour of spectral cluster (which connects to the eigenfunction concentration project). A spectral cluster groups eigenfunctions with similar eigenvalue (out to a  window width w) together and studies their bulk properties.