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

# 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,yVx,yV and suppose that vkkΓvkkΓ is an orthobasis. (ΓΓ denotes the index set, which could be finite or infinite.) Then x=kΓαkvkx=kΓαkvk and y=kΓβkvky=kΓβkvk, 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 2 x , y V = α , β 2
(2)

This is Plancherel's theorem. Parseval's theorem follows since x,xV=α,α2x,xV=α,α2 which implies that xV2=x22xV2=x22. Thus, an orthobasis makes every inner product space equivalent to 22!

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