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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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IV.11 Harmonic Analysis

Terence Tao

1 Introduction

Much of analysis tends to revolve around the study of general classes of FUNCTIONS [I.2 §2.2] and OPERATORS [III.50]. The functions are often real-valued or complex- valued, but may take values in other sets, such as a VECTOR SPACE [I.3 §2.3] or a MANIFOLD [I.3 §6.9]. An operator is itself a function, but at a “second level,” because its domain and range are themselves spaces of functions: that is, an operator takes a function (or perhaps more than one function) as its input and returns a transformed function as its output. Harmonic analysis focuses in particular on the quantitative properties of such functions, and how these quantitative properties change when various operators are applied ...

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