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Parseval's Theorem

Module by: Richard Baraniuk

Easy to show from plancharel that if BB is an ONB and x=Bα x B α then the l 2 l 2 norm/energy of x x is equivalent to l 2 l 2 norm/energy of α α:

Parseval's Theorem

∥x∥2=∥α∥2 2 x 2 α (1)

Note:

Parseval's theorem does not hold for arbitrary l p l p norm, only for p=2 p 2 .

Example 1

B B is an ONB.

x=Bα x B α , y=Bβ y B β

l 2 l 2 distance between x x and y y is ε.

What is the l 2 l 2 distance between α α and β β?

Example 2

Measuring error in JPEG image compression.

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