# 1.10 Signal approximation in a hilbert space

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

## The orthogonality principle

$\stackrel{^}{x}\in A$ is the minimizer of $∥x,-,\stackrel{^}{x}∥$ if any only if $\stackrel{^}{x}-x\perp A$ i.e., $〈\stackrel{^}{x},-,x,,,y〉=0$ for all $y\in A$ .

1. Suppose that $\stackrel{^}{x}-x\perp A$ . Then for any $y\in A$ with $y\ne \stackrel{^}{x},$ ${∥y,-,x∥}^{2}={∥y,-,\stackrel{^}{x},+,\stackrel{^}{x},-,x∥}^{2}$ . Note that $y-\stackrel{^}{x}\in A$ , but $\stackrel{^}{x}-x\perp A$ , so that $〈y,-,\stackrel{^}{x},,,\stackrel{^}{x},-,x〉=0$ , and we can apply Pythagoras to obtain ${∥y,-,x∥}^{2}={∥y,-,\stackrel{^}{x}∥}^{2}+∥\stackrel{^}{x},-,x∥$ . Since $y\ne \stackrel{^}{x}$ , we thus have that ${∥y,-,x∥}^{2}>{∥\stackrel{^}{x},-,x∥}^{2}$ . Thus $\stackrel{^}{x}$ must be the closest point in $A$ to $x$ .
2. Suppose that $\stackrel{^}{x}$ minimizes $∥x,-,\stackrel{^}{x}∥$ . Suppose for the sake of a contradiction that $\exists y\in A$ such that $∥y∥=1$ and $〈x,-,\stackrel{^}{x},,,y〉=\delta \ne 0$ .

Let $z=\stackrel{^}{x}+\delta y$ .

$\begin{array}{cc}\hfill {∥x,-,z∥}^{2}& ={∥x,-,\stackrel{^}{x},-,\delta ,y∥}^{2}\hfill \\ & =〈x,-,\stackrel{^}{x},,,x,-,\stackrel{^}{x}〉-〈x,-,\stackrel{^}{x},,,\delta ,y〉-〈\delta ,y,,,x,-,\stackrel{^}{x}〉+〈\delta ,y,,,\delta ,y〉\hfill \\ & ={∥x,-,\stackrel{^}{x}∥}^{2}-\overline{\delta }\delta -\delta \overline{\delta }+\delta \overline{\delta }\hfill \\ & ={∥x,-,\stackrel{^}{x}∥}^{2}-{|\delta |}^{2}.\hfill \end{array}$

Thus $∥x,-,z∥\le ∥x,-,\stackrel{^}{x}∥$ , contradicting the assumption that $\stackrel{^}{x}$ minimizes $∥x,-,\stackrel{^}{x}∥.$

This result suggests a that a possible method for finding the bestapproximation to a signal $x$ from a vector space $V$ is to simply look for a vector $\stackrel{^}{x}$ such that $\stackrel{^}{x}-x\perp V$ . 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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