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Parseval's and Plancherel's Theorems

Module by: Mark A. Davenport. E-mail the author

When dealing with transform coefficients, we will see that our notions of distance and angle carry over to the coefficient space.

Let x,y∈Vx,y∈V and suppose that vkk∈Γvkk∈Γ is an orthobasis. (ΓΓ denotes the index set, which could be finite or infinite.) Then x=∑k∈Γαkvkx=∑k∈Γαkvk and y=∑k∈Γβkvky=∑k∈Γβkvk, and

x , y V = ∑ k ∈ Γ α k β k ¯ . x , y V = ∑ k ∈ Γ α k β k ¯ .
(1)

So

x , y V = α , β ℓ 2 x , y V = α , β ℓ 2
(2)

This is Plancherel's theorem. Parseval's theorem follows since x,xV=α,αℓ2x,xV=α,αℓ2 which implies that xV2=xℓ22xV2=xℓ22. Thus, an orthobasis makes every inner product space equivalent to ℓ2ℓ2!

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