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The Wavelet Scaling Equation

Module by: Phil Schniter

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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 (1)
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 k∈ℤ k , we construct an orthonormal basis for W k W k and denote it by { ψ k , n t|n∈ℤ} 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=2-k2φ2-kt−n 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∈ ℒ 2 ψ t ℒ 2 : ψ k , n t=2-k2ψ2-kt−n . 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 W1⊂V0 W1 V0 , there must exist {gn|n∈ℤ} 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 (2)
⇔ 12ψ12t=∑n=-∞∞gnφt−n ⇔ 1 2 ψ 1 2 t n g n φ t n

Wavelet Scaling Equation

ψt=2∑n=-∞∞gnφ2t−n ψ t 2 n g n φ 2 t n (3)
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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