A Hierarchy of Detail in the Haar System 2.3 2003/01/17 2005/11/17 16:07:33.158 US/Central Phil Schniter schniter@ee.eng.ohio-state.edu Charlet Reedstrom charlet@rice.edu Phil Schniter schniter@ee.eng.ohio-state.edu Haar scaling function Given a mother scaling function φ t ℒ 2 — the choice of which will be discussed later — let us construct scaling functions at "coarseness-level-k" and "shift-n" as follows: φ k , n t 2 k 2 φ 2 k t n . Let us then use V k to denote the subspace defined by linear combinations of scaling functions at the k th level: V k span φ k , n t n . In the Haar system, for example, V 0 and V 1 consist of signals with the characteristics of x 0 t and x 1 t illustrated in .

We will be careful to choose a scaling function φ t which ensures that the following nesting property is satisfied: … V 2 V 1 V 0 V -1 V -2 … coarse   detailed In other words, any signal in V k can be constructed as a linear combination of more detailed signals in V k − 1 . (The Haar system gives proof that at least one such φ t exists.) The nesting property can be depicted using the set-theoretic diagram, , where V − 1 is represented by the contents of the largest egg (which includes the smaller two eggs), V 0 is represented by the contents of the medium-sized egg (which includes the smallest egg), and V 1 is represented by the contents of the smallest egg.

Going further, we will assume that φ t is designed to yield the following three important properties: φ k , n t n constitutes an orthonormal basis for V k , V ∞ 0 (contains no signals). While at first glance it might seem that V ∞ should contain non-zero constant signals (e.g., x t a for a ), the only constant signal in ℒ 2 , the space of square-integrable signals, is the zero signal. V − ∞ ℒ 2 (contains all signals). Because φ k , n t n is an orthonormal basis, the best (in ℒ 2 norm) approximation of x t ℒ 2 at coarseness-level-k is given by the orthogonal projection, x k t n c k , n φ k , n t c k , n φ k , n t x t

We will soon derive conditions on the scaling function φ t which ensure that the properties above are satisfied.