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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 V W k + 1 W k + 2 W k + 3 i = k + 1 W i
If we take the limit as k , we find that
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 2
This implies that, for any f t 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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