12.5 The wavelet scaling equation

The difference in detail between $V_k$ and $V_{k-1}$ will be described using $W_k$ , the orthogonal complement of $V_k$ in $V_{k-1}$ :

Suppose now that, for each $k\in \mathbb{Z}$ , we construct an orthonormal basis for $W_k$ and denote it by $\{{\psi }_{k,n}(t)\colon n\in \mathbb{Z}\}$ . It turns out that, because every $V_k$ has a basis constructed from shifts and stretches of a mother scaling function ( i.e. , ${\phi }_{k,n}(t)=2^{-\left(\frac{k}{2}\right)}\phi (2^{-k}t-n)$ , every $W_k$ has a basis that can be constructed from shifts and stretches of a "mother wavelet" $\psi (t)\in {ℒ}_2$ : $${\psi }_{k,n}(t)=2^{-\left(\frac{k}{2}\right)}\psi (2^{-k}t-n)\text{.}$$ The Haar system will soon provide us with a concrete example.

Let's focus, for the moment, on the specific case $k=1$ . Since $W_1\subset V_0$ , there must exist $\{g(n)\colon n\in \mathbb{Z}\}$ such that: