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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 of point 'p' and line 'l' mentioned above.

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

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

Calculating α

How to calculate α ^ ?

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

We can do the same thing in higher dimensions.

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

  1. Find an orthonormal basis b 1 b k for V
  2. Project x onto V using y i 1 k x b i b i then y is the closest point in V to x and (x-y) ⊥ V ( v v V x y v 0

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x 3 , V span 1 0 0 0 1 0 , x a b c . So, y i 1 2 x b i b i a 1 0 0 b 0 1 0 a b 0

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V = {space of periodic signals with frequency no greater than 3 w 0 }. Given periodic f(t), what is the signal in V that best approximates f?

  1. { 1 T w 0 k t , k = -3, -2, ..., 2, 3} is an ONB for V
  2. g t 1 T k -3 3 f t w 0 k t w 0 k t is the closest signal in V to f(t) ⇒ reconstruct f(t) using only 7 termsof its Fourier series .

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Let V = {functions piecewise constant between the integers}

  1. ONB for V.

b i 1 i 1 t i 0 where { b i } is an ONB.

Best piecewise constant approximation? g t i f b i b i f b i t f t b i t t i 1 i f t

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

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Source:  OpenStax, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15
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