# 7.2 A hierarchy of detail in the haar system

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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: $\dots \subset {V}_{2}\subset {V}_{1}\subset {V}_{0}\subset {V}_{-1}\subset {V}_{-2}\subset \dots$ $\text{coarse}\text{}\text{detailed}$ 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).
Because $\{{\phi }_{k,n}(t)\colon n\in \mathbb{Z}\}$ is an orthonormal basis, the best (in ${ℒ}_{2}$ norm) approximation of $x(t)\in {ℒ}_{2}$ at coarseness-level- $k$ is given by the orthogonal projection,
${x}_{k}(t)=\sum_{n=()}$ c k , n φ k , n t
${c}_{k,n}={\phi }_{k,n}(t)\dot x(t)$

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

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