As an application of Parseval's Theorem, say vkk=1∞vkk=1∞ is an orthobasis for an
inner product space of VV.
Let AA be the subspace spanned by the first 10 elements of vkvk, i.e., A= span v1,...,v10A= span v1,...,v10
- Given x∈vx∈v, what is the closest point in AA (call it x^x^) to
xx? We have seen that it is x^=∑k=110x,vkvkx^=∑k=110x,vkvk
- How good of an approximation is x^x^ to xx? Measured with ∥·∥V∥·∥V:
∥x-x^∥V2=∑k>10〈x,vk〉vkV2=∑k>10|〈x,vk〉|2∥x-x^∥V2=∑k>10〈x,vk〉vkV2=∑k>10|〈x,vk〉|2
(1)
Since we also have that ∥x∥V2=∑k=1∞|〈x,vk〉|2∥x∥V2=∑k=1∞|〈x,vk〉|2, the approximation x^x^ will be “good” if
the first 10 transform coefficients contain “most” of the total energy.
Constructing these types of approximations is exactly what is done in image
compression.
