Senior Honors Thesis Presentation: Multiresolution Analysis for Singular Integrals: One-Parameter, Product, and Entangled Settings

Abstract: This is an introduction to the study of singular integrals via dyadic multiresolution analysis. The idea is to produce “representation theorems” that exhibit singular integrals as sums of simpler dyadic operators. These methods are advantageous in part because they are adaptable to various settings. Here we look at three settings: the one-parameter case where we view $\R^d$ as a one-parameter space, the multi-parameter case where we view $\R^d$ as a product of lower-dimensional spaces, and the Zygmund case that lies somewhere between the other two. We highlight how the multiresolution analysis needs to be changed in each setting. 

Faculty Advisor: Henri Martikainen