A case study: Duality for Cantor sets

Abstract

A well-known principle in Fourier series (reviewed in Section 3.1) for functions on a finite interval states that an orthogonal trigonometric basis exists and will be indexed by an arithmetic progression of (Fourier) frequencies, i.e., by integers times the inverse wave length. Similarly, in higher dimensions d, we define periodicity in terms of a lattice of rank d. The principle states that for d-periodic functions on ℝ^d, the appropriate Fourier frequencies may then be realized by a certain dual rank-d lattice. In this case, the inverse relation is formulated as a duality principle for lattices; see, for example, [JoPe93] for a survey of this point.

Closer to this book and equally illuminating are the many problems trigered by a sound or a picture. Only afterwards is a formula devised, and then proclamed.⋯ —Benoit B. Mandelbrot