Infinite products

Abstract

In the first two chapters, we introduced the random walks that are used throughout the book. We outlined this in the context of endomorphisms of compact spaces X and combinatorial trees; and we showed in Chapters 3 and 4 how this applies to wavelets and fractals. The combinatorial trees to keep in mind for illustration are sketched in Figures 1.1 (p. 8) and 2.1 (the Farey tree, p. 42). Recall further that the transition probabilities in the random-walk model are assigned via a prescribed function W on X which is assumed to satisfy a certain normalization condition. Within the context of signals, W is the absolute square of some frequency function m, or of a wavelet filter. The various paths within our tree can originate at points x chosen from the set X. As before, X carries a fixed finite-to-one endomorphism σ. If x and y are points in X such that σ (y)=x, then the number W (y) represents the probability of a transition from x to y. Step-by-step conditional probabilities and finite products are used in assigning probabilities to finite paths which originate at x. (The simplest instance of this idea is for the case when X is the circle, i.e., the one-torus $\mathbb{T}$. For each N, we may then consider σ (z):=z^N. And in the context of wavelet constructions, we introduced the additive formulation of the distinct branches of the inverse of z →z^N when z is complex and restricted to $\mathbb{T}$.)

Albert Einstein to Oscar Veblen (a math professor): “The Lord God is subtle, but malicious he is not.” (Raffiniert ist der Herr Gott, aber boshaft ist Er nicht.) And ten years later: “I have second thoughts. Maybe God is malicious.” —Albert Einstein