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

# Signal Approximation in a Hilbert Space

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

We will now revisit “The Fundamental Theorem of Approximation” for the extremely important case where our set AA is a subspace. Specifically, suppose that HH is a Hilbert space, and let AA be a (closed) subspace of HH. From before, we have that for any xHxH there is a unique x^Ax^A such that x^x^ is the closest point in AA to xx. When AA is also a subspace, we also have:

## Theorem 1: The Orthogonality Principle

x^Ax^A is the minimizer of x-x^x-x^ if any only if x^-xAx^-xA i.e., x^-x,y=0x^-x,y=0 for all yAyA.

### Proof

1. (a) Suppose that x^-xAx^-xA. Then for any yAyA with yx^,yx^,y-x2=y-x^+x^-x2y-x2=y-x^+x^-x2. Note that y-x^Ay-x^A, but x^-xAx^-xA, so that y-x^,x^-x=0y-x^,x^-x=0, and we can apply Pythagoras to obtain y-x2=y-x^2+x^-xy-x2=y-x^2+x^-x. Since yx^yx^, we thus have that y-x2>x^-x2y-x2>x^-x2. Thus x^x^ must be the closest point in AA to xx.
2. (b) Suppose that x^x^ minimizes x-x^x-x^. Suppose for the sake of a contradiction that yAyA such that y=1y=1 and x-x^,y=δ0x-x^,y=δ0.

Let z=x^+δyz=x^+δy.

x - z 2 = x - x ^ - δ y 2 = x - x ^ , x - x ^ - x - x ^ , δ y - δ y , x - x ^ + δ y , δ y = x - x ^ 2 - δ ¯ δ - δ δ ¯ + δ δ ¯ = x - x ^ 2 - | δ | 2 . x - z 2 = x - x ^ - δ y 2 = x - x ^ , x - x ^ - x - x ^ , δ y - δ y , x - x ^ + δ y , δ y = x - x ^ 2 - δ ¯ δ - δ δ ¯ + δ δ ¯ = x - x ^ 2 - | δ | 2 .
(1)

Thus x-zx-x^x-zx-x^, contradicting the assumption that x^x^ minimizes x-x^.x-x^.

This result suggests a that a possible method for finding the best approximation to a signal xx from a vector space VV is to simply look for a vector x^x^ such that x^-xVx^-xV. In the coming lectures we will show how to do this, but it will require a brief review of some concepts from linear algebra.

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