Szego Seminar: A friendly overview of dyadic harmonic analysis.

Abstract: The word “harmonic” comes from the Greek word “harmonikos”, that literally means “skilled in music”. Sorry to disappoint you: I won’t talk about music this time, but it can be a pleasant way to think of the main subject of this talk: waves and signals. 

A superposition of simple waves such as cosines and sines, that oscillate with different phases, amplitudes and periods is called harmonic function. Motivated by the enormous number of applications, the goal of modern analysis is to decompose any reasonable function as a sum of simpler functions that are easy to understand.  

In other words, the essence of harmonic analysis is laziness: we aim to find an easy way to deal with complicated objects. Well… easier said than done!

Not satisfied with cosines and sines, that periodically repeat over time, people started to study an even simpler class of signals, called “wavelets”, which are wave-type compactly supported functions: dyadic harmonic analysis is related to a specific class of wavelets, called “Haar wavelets”, that are basically the easiest thing that we can think of, as we will see. 

For a long time, dyadic models have been considered as toy models to understand more complicated objects, called Calderón-Zygmund operators. What if they are enough? Can we describe somehow any such operator only using dyadic models? We will try to answer this question.

Host: Pooja Joshi