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# Error of the Best Approximation in an Orthobasis

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

As an application of Parseval's Theorem, say vkk=1vkk=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

1. Given xvxv, 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
2. How good of an approximation is x^x^ to xx? Measured with ·V·V:
x-x^V2=k>10x,vkvkV2=k>10|x,vk|2x-x^V2=k>10x,vkvkV2=k>10|x,vk|2
(1)

Since we also have that xV2=k=1|x,vk|2xV2=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.

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