Mise en Scène: Mainly a Historical Perspective

A recurring construct in previous chapters was based on this simple blueprint:

At the very outset, if nothing is known or assumed about the ‘building blocks’ x1,…,xn, then their product x1⊗…⊗xn is merely a formal object, and not much more can be said. If something is known about E1,…,En, then meaning could be ascribed to x1⊗ … ⊗xn, and analysis of E1 ⊗…⊗En would proceed accordingly. In our specific context, we considered Rademacher functions and their products. We considered the set of independent functions R = {rk} on Ω = {−1, 1}ℕ, and viewed the elements in the n-fold R⊗…⊗R as functions on Ωn. An underlying theme has been that Rademacher functions are basic objects from which all else is constructed, a notion that can be formulated effectively in a framework of harmonic analysis. And that is our purpose in this chapter: to learn and analyze this framework, as it is built from the ground up.

Loosely put, harmonic analysis is about representing general phenomena in terms of familiar phenomena. The subject's beginnings – in the mid-eighteenth century, about ninety years after the invention of the calculus – were rooted in the notion that arbitrary functions could be represented by series of sines and cosines.