Skip to content Skip to navigation

OpenStax-CNX

You are here: Home » Content » Approximation and Projections in Hilbert Space

Navigation

Recently Viewed

This feature requires Javascript to be enabled.

Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.
 

Approximation and Projections in Hilbert Space

Module by: Justin Romberg. E-mail the author

Summary: This module introduces approximation and projections in Hilbert space.

Note: You are viewing an old version of this document. The latest version is available here.

Introduction

Given a line 'l' and a point 'p' in the plane, what's the closest point 'm' to 'p' on 'l'?

Figure 1: Figure of point 'p' and line 'l' mentioned above.
Figure 1 (approx_f1.png)

Same problem: Let xx and vv be vectors in R2 2 . Say v=1 v 1 . For what value of αα is xαv 2 x α v 2 minimized? (what point in span{v} best approximates xx?)

Figure 2:
Figure 2 (approx_f2.png)

The condition is that x α ^ v x α ^ v and αv α v are orthogonal.

Calculating α

How to calculate α ^ α ^ ?

We know that ( x α ^ v x α ^ v ) is perpendicular to every vector in span{v}, so β,β:x α ^ v,βv=0 β β x α ^ v β v 0 β¯(x,v) α ^ β¯(v,v)=0 β x v α ^ β v v 0 because v,v=1 v v 1 , so ((x,v) α ^ =0)( α ^ =x,v) x v α ^ 0 α ^ x v Closest vector in span{v} = (x,v)v x v v , where (x,v)v x v v is the projection of xx onto vv.

Point to a plane?

Figure 3:
Figure 3 (approx_f3.png)

We can do the same thing in higher dimensions.

Exercise 1

Let VH V H be a subspace of a Hilbert space H. Let xH x H be given. Find the yV y V that best approximates xx. i.e., xy x y is minimized.

Solution

  1. Find an orthonormal basis b1bk b1 bk for VV
  2. Project xx onto VV using y=i=1k(x,bi)bi y i 1 k x bi bi then yy is the closest point in V to x and (x-y) ⊥ V ( v,vV:xy,v=0 v v V x y v 0

Example 1

xR3 x 3 , V=span( 1 0 0 )( 0 1 0 ) V span 1 0 0 0 1 0 , x=( a b c ) x a b c . So, y=i=12(x,bi)bi=a( 1 0 0 )+b( 0 1 0 )=( a b 0 ) y i 1 2 x bi bi a 1 0 0 b 0 1 0 a b 0

Example 2

V = {space of periodic signals with frequency no greater than 3 w0 3 w0 }. Given periodic f(t), what is the signal in V that best approximates f?

  1. { 1Tei w0 kt 1 T w0 k t , k = -3, -2, ..., 2, 3} is an ONB for V
  2. gt=1Tk=-33(ft,ei w0 kt)ei w0 kt g t 1 T k -3 3 f t w0 k t w0 k t is the closest signal in V to f(t) ⇒ reconstruct f(t) using only 7 terms of its Fourier series.

Example 3

Let V = {functions piecewise constant between the integers}

  1. ONB for V.

bi ={1  if  i1t<i0  otherwise   bi 1 i 1 t i 0 where {bibi} is an ONB.

Best piecewise constant approximation? gt=i=(f, bi ) bi g t i f bi bi f, bi =ft bi tdt=i1iftdt f bi t f t bi t t i 1 i f t

Example 4

This demonstration explores approximation using a Fourier basis and a Haar Wavelet basis.See here for instructions on how to use the demo.

LabVIEW Example: (run) (source)

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

Lenses

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