Skip to content Skip to navigation


You are here: Home » Content » Hilbert Spaces in Signal Processing


Recently Viewed

This feature requires Javascript to be enabled.

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,yWx,yW and λ(0,1)λ(0,1), λx+(1-λ)yWλx+(1-λ)yW.

Theorem 1: The Fundamental Theorem of Approximation

Let AA be a nonempty, closed (complete), convex set in a Hilbert space HH. For any xHxH 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.

Content actions

Download module as:

Add module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens


A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks