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Hilbert Spaces in Signal Processing

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

What makes Hilbert spaces so useful in signal processing? In modern signal processing, we often represent a signal as a point in high-dimensional space. Hilbert spaces are spaces in which our geometry intuition from R3R3 is most trustworthy. As an example, we will consider the approximation problem.

Definition 1.

A subset WW of a vector space VV is convex if for all x,y∈Wx,y∈W and λ∈(0,1)λ∈(0,1), λx+(1-λ)y∈Wλx+(1-λ)y∈W.

Theorem 1: The Fundamental Theorem of Approximation

Let AA be a nonempty, closed (complete), convex set in a Hilbert space HH. For any x∈Hx∈H there is a unique point in AA that is closest to xx, i.e., xx has a unique “best approximation” in AA.

Figure 1: The best approximation to xx in convex set AA.
An illustration showing a convex set A and a point x that lies outside this set. The closest point to x in A is x_hat.

Note that in non-Hilbert spaces, this may not be true! The proof is rather technical. See Young Chapter 3 or Moon and Stirling Chapter 2. Also known as the “closest point property”, this is very useful in compression and denoising.

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