The Central Idea: The Hilbert Transform

Abstract

The Hilbert transform is, without question, the most important operator in analysis. It arises in many different contexts, and all these contexts are intertwined in profound and influential ways. What it all comes down to is that there is only one singular integral in dimension 1, and it is the Hilbert transform. The philosophy is that all significant analytic questions reduce to a singular integral; and in the first dimension there is just one choice.