Infinite series are delicate beasts that have to be summed with tender love and care. To describe a one-dimensional signal, we can steadily pile on higher and higher frequency waves to approximate it with any accuracy. But the moment we step up a dimension, and consider a “signal” in a square region (e.g. colours at the pixel locations over a square image), we have a problem: how do we add “higher frequencies” if they are described by two components? The question of how to gather up the terms of two-variable Fourier series in a way that gives us sensible answers is a deep and intricate one, touching on various areas of mathematics from operator theory, to geometry, to the theory of numbers.
In this talk we will outline this classical (but now somewhat neglected) problem in an accessible manner, and present our recent results on the way to generalising the problem to “Fourier” (or eigenfunction) expansions in other planar domains (e.g. images in triangular or circular regions). This is joint work with Prof. James C. Robinson.
Ryan Acosta Babb is final year PhD student at the University of Warwick, where he researches the Lp convergence of eigenfunction expansions for second-order linear differential operators in the plane. He previously obtained (from the same institution) a Masters of Advanced Studies in Mathematics, with a dissertation on scaling limits of the Gaussian Free Field, and an undergraduate degree in Mathematics and Philosophy with a slightly different focus in my final-year thesis: Does Kant’s ethics presuppose belief in God? When not thinking about Fourier series or debugging LaTeX, Ryan enjoys studying ancient languages such as Greek and Latin.