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Approximation and Projections in Hilbert Space

Module by: Justin Romberg. E-mail the author

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

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 ∥x−α⁢v∥ 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 α⁢v 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 〈x− α ^ ⁢v,β⁢v〉=0  ,   ∀⁢β    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.

We can do the same thing in higher dimensions.

Exercise 1

Let V⊂H V H be a subspace of a Hilbert space H. Let x∈H x H be given. Find the y∈V y V that best approximates xx. i.e., ∥x−y∥ x y is minimized.

Solution

  1. Find an orthonormal basis b1…bk 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 ( 〈x−y,v〉=0  ,   ∀⁢v∈V    v ∀ v V x y v 0

Example 1

x∈R3 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. { 1T⁢ej⁢ w0 ⁢k⁢t 1 T w0 k t , k = -3, -2, ..., 2, 3} is an ONB for V
  2. g⁢t=1T⁢∑k=-33〈(f⁢t,ej⁢ w0 ⁢k⁢t)〉⁢ej⁢ w0 ⁢k⁢t 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  i−1≤t<i0  otherwise   bi 1 i 1 t i 0 where {bibi} is an ONB.

Best piecewise constant approximation? g⁢t=∑i=−∞∞〈(f, bi )〉⁢ bi g t i f bi bi 〈f, bi 〉=∫−∞∞f⁢t⁢ bi ⁢tdt=∫i−1if⁢tdt 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)

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