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Course by: Janko Calic. E-mail the author

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

Module by: Phil Schniter. E-mail the author

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 V = 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 V 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
2 = V = i = W i 2 V i = W i
(2)
Moreover,
(W1V1)( W k 2 V 1 )(W1 W k 2 ) W1 V1 W k 2 V 1 W1 W k 2
(3)
(W2V2)( 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 nZ ψ k , n t n form an orthonormal basis for W k W k , the results above imply that
ψ k , n t nkZ constitutes an orthonormal basis for 2 ψ k , n t n k constitutes an orthonormal basis for 2
(6)
This implies that, for any ft 2 f t 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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