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7.8 Wavelets: a countable orthonormal basis for the space of square

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Recall that V k W k + 1 V k + 1 and that 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
If we take the limit as k ∞ , we find that
ℒ 2 V − ∞ ⊕ i = − ∞ ∞ W i
Moreover,
⊥ W 1 V 1 W k ≥ 2 V 1 ⊥ W 1 W k ≥ 2
⊥ W 2 V 2 W k ≥ 3 V 2 ⊥ W 2 W k ≥ 3
from which it follows that
⊥ W k W j ≠ k
or, in other words, all subspaces W k are orthogonal to one another. Since the functions ψ k , n t n form an orthonormal basis for W k , the results above imply that
ψ k , n t n k constitutes an orthonormal basis for ℒ 2
This implies that, for any f t ℒ 2 , we can write
f t k ∞ ∞ m ∞ ∞ d k m ψ k , m t
d k m ψ k , m t f t
This is the key idea behind the orthogonal wavelet system that we have been developing!
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Source:  OpenStax, Digital signal processing (ohio state ee700). OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10144/1.8
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