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The Wavelet Scaling Equation

Module by: Phil Schniter. E-mail the author

The difference in detail between V k V k and V k 1 V k 1 will be described using W k W k , the orthogonal complement of V k V k in V k 1 V k 1 :

V k 1 = V k W k V k 1 V k W k
At times it will be convenient to write W k = V k W k V k . This concept is illustrated in the set-theoretic diagram, Figure 1.

Figure 1
Figure 1 (wavelet_scaling_set.png)

Suppose now that, for each kZ k , we construct an orthonormal basis for W k W k and denote it by ψ k , n t nZ ψ k , n t n . It turns out that, because every V k V k has a basis constructed from shifts and stretches of a mother scaling function (i.e., φ k , n t=2k2φ2ktn φ k , n t 2 k 2 φ 2 k t n , every W k W k has a basis that can be constructed from shifts and stretches of a "mother wavelet" ψt 2 ψ t 2 : ψ k , n t=2k2ψ2ktn . ψ k , n t 2 k 2 ψ 2 k t n . The Haar system will soon provide us with a concrete example .

Let's focus, for the moment, on the specific case k=1 k 1 . Since W1V0 W1 V0 , there must exist gn nZ g n n such that:

ψ 1 , 0 t=n=gn φ 0 , n t ψ 1 , 0 t n g n φ 0 , n t
12ψ12t=n=gnφtn 1 2 ψ 1 2 t n g n φ t n

Wavelet Scaling Equation

ψt=2n=gnφ2tn ψ t 2 n g n φ 2 t n
To be a valid scaling-function/wavelet pair, φt φ t and ψt ψ t must obey the wavelet scaling equation for some coefficient set gn g n .

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