Harmonic Analysis

Webster's dictionary defines the word analysis as a breaking up of a whole into its parts as to find out their nature. This is indicative of one of the most fundamental principles manifesting itself in Harmonic Analysis, having to do with ecomposing a mathematical object, such as a function/distribution, or an operator, into simpler entities (enjoying certain specialized localization, cancellation, and size conditions), analyzing these smaller pieces individually, and then organizing this local information in a global, coherent manner, in order to derive conclusions about the original object of study. This principle goes back at least as far as the ground breaking work of J. Fourier in the early 1800's who had the vision of using superposition of sine and cosine graphs (with various amplitudes) as a means of creating the shape of the graph of a relatively arbitrary function. In such a scenario, the challenge is to create a dictionary between the nature of the Fourier coefficients on the one hand, and the functional-analytic properties of the original function, such as membership to various Lebesgue spaces, on the other hand.

This point of view has received further impetus through the development of Littlewood-Paley theory, leading up to the modern theory of function spaces of Triebel-Lizorkin and Besov type.  Another embodiment of the pioneering ideas of Fourier that has fundamentally shaped present day Harmonic Analysis is the theory of Hardy spaces viewed through the perspective of atomic and molecular techniques. In this context, the so-called atoms and molecules play the role of the sine and cosine building blocks (though they only constitute an ``overdetermined basis" as opposed to a genuine linear basis).

From its original roots, the field of Harmonic Analysis has presently grown into a  vast and intricate collection of results and techniques with deep implications in such diverse branches of mathematics such as Partial Differential Equations, Complex Analysis, Fourier Analysis, Singular Integrals, Geometric Measure Theory, Nonlinear Potential Theory, Numerical Analysis, Approximation Theory, Wavelets, Combinatorics, Number Theory, etc.