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Wavelets: A Countable Orthonormal Basis for the Space of Square-Integrable Functions

Module by: Phil Schniter

Recall that V k = W k + 1 ⊕ V k + 1 V k W k + 1 V k + 1 and that V k + 1 = W k + 2 ⊕ V k + 2 V k + 1 W k + 2 V k + 2 . Putting these together and extending the idea yields
V k = W k + 1 ⊕ W k + 2 ⊕ V k + 2 = W k + 1 ⊕ W k + 2 ⊕…⊕ W ℓ ⊕ V ℓ = W k + 1 ⊕ W k + 2 ⊕ W k + 3 ⊕…= ⊕ i = k + 1 ∞ W i V k W k + 1 W k + 2 V k + 2 W k + 1 W k + 2 … W ℓ V ℓ W k + 1 W k + 2 W k + 3 … ⊕ i = k + 1 ∞ W i (1)
If we take the limit as k→-∞ k , we find that
ℒ 2 = V − ∞ = ⊕ i = − ∞ ∞ W i ℒ 2 V − ∞ ⊕ i = − ∞ ∞ W i (2)
Moreover,
W1⊥V1∧ W k ≥ 2 ⊂ V 1 ⇒W1⊥ W k ≥ 2 ⊥ W1 V1 W k ≥ 2 V 1 ⊥ W1 W k ≥ 2 (3)
W2⊥V2∧ W k ≥ 3 ⊂V2⇒W2⊥ W k ≥ 3 ⊥ W2 V2 W k ≥ 3 V2 ⊥ W2 W k ≥ 3 (4)
from which it follows that
Wk⊥ W j ≠ k ⊥ Wk W j ≠ k (5)
or, in other words, all subspaces W k W k are orthogonal to one another. Since the functions { ψ k , n t|n∈ℤ} ψ k , n t n form an orthonormal basis for W k W k , the results above imply that
{ ψ k , n t|n∧k∈ℤ} constitutes an orthonormal basis for ℒ 2 ψ k , n t n k constitutes an orthonormal basis for ℒ 2 (6)
This implies that, for any ft∈ ℒ 2 f t ℒ 2 , we can write
ft=∑k=-∞∞∑m=-∞∞ d k m ψ k , m t f t k m d k m ψ k , m t (7)
d k m=< ψ k , m t,ft> d k m ψ k , m t f t (8)
This is the key idea behind the orthogonal wavelet system that we have been developing!

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