Mathematics Colloquium: How Much Can a Function Be Concentrated in Space and Frequency? From Fourier Uncertainty to Eigenvalue Distributions and Geometry

Speaker:  Azita Mayeli, CUNY 

Title:  How Much Can a Function Be Concentrated in Space and Frequency? From Fourier Uncertainty to Eigenvalue Distributions and Geometry

Abstract:  A fundamental principle of Fourier analysis tells us that a function and its Fourier transform cannot both be perfectly localized. Nevertheless, a function can be highly concentrated in a given spatial region while its Fourier transform is concentrated in a prescribed frequency region. This raises two natural questions: How well can a function be concentrated in this way, and how many independent functions can achieve such concentration?

These questions go back to the classical work of Slepian, Landau, and Pollak and lead to an interesting phenomenon: the eigenvalues of certain operators cluster near 0 and 1, with relatively few in between.

In this talk, I will introduce these ideas through simple examples and explain how the geometry of the spatial and frequency regions influences the eigenvalue distribution. I will discuss recent joint work with Kevin Hughes and Arie Israel on eigenvalue estimates for irregular domains, as well as related work with Luis Gomez and Hana Huber on discrete multiband concentration problems.

No prior knowledge of concentration operators will be assumed.

Time:  Friday, November 6, 3:30pm – 4:30pm

Place:  Exploratory Hall, room 4106 or Zoom