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Computing the Best Approximation

Module by: Mark A. Davenport. E-mail the author

Recall that if PP is an orthogonal projection onto a subspace AA, we can write any xx as

x = P x + ( I - P ) x x = P x + ( I - P ) x
(1)

where PxAPxA and (I-P)xA(I-P)xA. We now turn to how to actually find PP.

We begin with the finite-dimensional case, assuming that {v1,...,vN}{v1,...,vN} is a basis for AA. If (I-P)xA(I-P)xA then we have that for any xx

( I - P ) x , v j = 0 for j = 1 , . . . , N ( I - P ) x , v j = 0 for j = 1 , . . . , N
(2)

We also note that since PxAPxA, we can write Px=k=1NckvkPx=k=1Nckvk. Thus we obtain

x - k = 1 N c k v k , v j = 0 for j = 1 , . . . , N x - k = 1 N c k v k , v j = 0 for j = 1 , . . . , N
(3)

from which we obtain

x , v j = k = 1 N c k v k , v j for j = 1 , . . . , N x , v j = k = 1 N c k v k , v j for j = 1 , . . . , N
(4)

We know xx and v1,...,vNv1,...,vN. Our goal is to find c1,...,cNc1,...,cN. Note that a procedure for calculating c1,...,ckc1,...,ck for any given xx is equivalent to one that computes PxPx.

To find c1,...,cNc1,...,cN, observe that Equation 4 represents a set of NN equations with NN unknowns.

v 1 , v 1 v 2 , v 1 v N , v 1 v 1 , v 2 v 2 , v 2 v N , v 2 v 1 , v N v 2 , v N v N , v N c 1 c 2 c N = x , v 1 x , v 2 x , v N v 1 , v 1 v 2 , v 1 v N , v 1 v 1 , v 2 v 2 , v 2 v N , v 2 v 1 , v N v 2 , v N v N , v N c 1 c 2 c N = x , v 1 x , v 2 x , v N
(5)

More compactly, we want to find a vector cCNcCN such that Gc=bGc=b where

b = x , v 1 x , v 2 x , v N b = x , v 1 x , v 2 x , v N
(6)

Note:

  • GG is called the “Grammian” or “Gram matrix” of {vj}{vj}
  • One can show since v1,...,vNv1,...,vN are linearly independent that GG is positive definite, and hence invertible.
  • Also note that by construction, GG is conjugate symmetric, or “Hermitian”, i.e., G=GHG=GH, where HH denotes the conjugate transpose of GG.

Thus, since G-1G-1 exists, we can write c=G-1bc=G-1b to calculate cc.

As a special case, suppose now that {vj}{vj} is an orthobasis for AA? What is GG? It is just the identity matrix II! Computing cc just got much easier, since now c=bc=b. Plugging this cc back into out formula for PxPx we obtain

P x = k = 1 N x , v k v k P x = k = 1 N x , v k v k
(7)

Just to verify, note that PP is indeed a projection matrix:

P ( P x ) = N k = 1 N j = 1 x , v j v j , v k v k = N k = 1 N j = 1 x , v j v j , v k v k = N j = 1 x , v j v j = P x . P ( P x ) = N k = 1 N j = 1 x , v j v j , v k v k = N k = 1 N j = 1 x , v j v j , v k v k = N j = 1 x , v j v j = P x .
(8)

Example Suppose fL2([0,4])fL2([0,4]) is given by

Example 1

Suppose fL2([0,4])fL2([0,4]) is given by

f ( t ) = t if t [ 0 , 1 2 ] 1 - t if t [ 1 2 , 1 ] . f ( t ) = t if t [ 0 , 1 2 ] 1 - t if t [ 1 2 , 1 ] .
(9)
Figure 1
Illustration of the function f(t).

Let A={ piecewise constant functions on [0,14),[14,12),[12,34),[34,1]}A={ piecewise constant functions on [0,14),[14,12),[12,34),[34,1]}. Our goal is to find the closest (in L2L2) function in AA to f(t)f(t). Using v1,...,v4v1,...,v4 from before, we can calculate c1=14c1=14, c2=0c2=0, c3=-216c3=-216, c4=216c4=216. Thus, we have that

f ^ ( t ) = 1 4 v 1 - 2 16 v 3 + 2 16 v 4 . f ^ ( t ) = 1 4 v 1 - 2 16 v 3 + 2 16 v 4 .
(10)
Figure 2
Illustration of the function f_hat(t), i.e., the best piecewise-constant approximation to f(t).

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