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Given a mother scaling function $\phi (t)\in {\mathcal{L}}_{2}$ —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:
We will soon derive conditions on the scaling function $\phi (t)$ which ensure that the properties above are satisfied.
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