12.8 Wavelets: a countable orthonormal basis for the space of

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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} … \oplus_{i = k + 1}^{\infty} W_{i}

If we take the limit as

k

∞ , we find that

ℒ_{2} V_{- \infty} \oplus_{i = - \infty}^{\infty} W_{i}

Moreover,

⊥ W_{1} V_{1} W_{k \geq 2} V_{1} ⊥ W_{1} W_{k \geq 2}

⊥ W_{2} V_{2} W_{k \geq 3} V_{2} ⊥ W_{2} W_{k \geq 3}

from which it follows that

⊥ W_{k} W_{j \neq 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, Dspa. OpenStax CNX. May 18, 2010 Download for free at http://cnx.org/content/col10599/1.5

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