Wavelets: A Countable Orthonormal Basis for the Space of Square-Integrable Functions 2.2 2003/01/17 2005/10/18 16:38:27.815 GMT-5 Phil Schniter schniter@ee.eng.ohio-state.edu Charlet Reedstrom charlet@rice.edu Phil Schniter schniter@ee.eng.ohio-state.edu orthogonal wavelets 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, ⊥ W1 V1 W k ≥ 2 V 1 ⊥ W1 W k ≥ 2 ⊥ W2 V2 W k ≥ 3 V2 ⊥ W2 W k ≥ 3 from which it follows that ⊥ Wk 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!