1.7 Hilbert spaces in signal processing

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 ${\mathbb{R}}^3$ is most trustworthy. As an example, we will consider the approximation problem.

Definition 1.

The fundamental theorem of approximation

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

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.