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Given a mother scaling function —the choice of which will be discussed later—let us construct scaling functions at"coarseness-level- and "shift- " as follows: Let us then use to denote the subspace defined by linear combinations of scaling functions at the level: In the Haar system, for example, and consist of signals with the characteristics of and illustrated in .
We will be careful to choose a scaling function which ensures that the following nesting property is satisfied: In other words, any signal in can be constructed as a linear combination of more detailed signals in . (The Haar system gives proof that at least one such exists.)
The nesting property can be depicted using the set-theoretic diagram, , where is represented by the contents of the largest egg (which includes the smaller two eggs), is represented by the contents of the medium-sized egg (which includes the smallest egg), and is represented by the contents of the smallest egg.
Going further, we will assume that is designed to yield the following three important properties:
We will soon derive conditions on the scaling function which ensure that the properties above are satisfied.
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