12.2 A hierarchy of detail in the haar system

Given a mother scaling function $\phi (t)\in {ℒ}_2$ —the choice of which will be discussed later—let us construct scaling functions at"coarseness-level- $\mathrm{k"}$ and "shift- $n$ " as follows: $${\phi }_{k,n}(t)=2^{-\left(\frac{k}{2}\right)}\phi (2^{-k}t-n)\text{.}$$ Let us then use $V_k$ to denote the subspace defined by linear combinations of scaling functions at the $k^{\mathrm{th}}$ level: $$V_k=\mathrm{span}(\{{\phi }_{k,n}(t)\colon n\in \mathbb{Z}\})\text{.}$$ 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 $\phi (t)$ which ensures that the following nesting property is satisfied: $$\dots \subset V_2\subset V_1\subset V_0\subset V_{-1}\subset V_{-2}\subset \dots$$ $$\text{coarse}\text{detailed}$$ 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 $\phi (t)$ 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 $\phi (t)$ is designed to yield the following three important properties:

• $\{{\phi }_{k,n}(t)\colon n\in \mathbb{Z}\}$ constitutes an orthonormal basis for $V_k$ ,
• $V_{\infty }=\{0\}$ (contains no signals).
  While at first glance it might seem that $V_{\infty }$ should contain non-zero constant signals ( e.g. , $x(t)=a$ for $a\in \mathbb{R}$ ), the only constant signal in ${ℒ}_2$ , the space of square-integrable signals, is the zero signal.
• $V_{-\infty }={ℒ}_2$ (contains all signals).

We will soon derive conditions on the scaling function $\phi (t)$ which ensure that the properties above are satisfied.