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Unformatted text preview: Chapter 4 Representation and Approximation 4.1 Best Approximation Suppose W is a subspace of a Banach space V . For any v V , consider the problem of nding a vector w W such that v w is as small as possible. Defnition 4.1.1. The vector w W is a best approximation of v V by vectors in W if v w v w for all w W . If W is spanned by the vectors w 1 ,...,w n V , then we can write v = w + e = s 1 w 1 + + s n w n + e , where e is the approximation error. This problem is, in general, rather difcult. However, if the norm cor responds to the induced norm of an inner product, then one can use orthogonal projection and the problem is greatly simplied. This chapter focuses mainly on computing the best approximation of arbitrary vectors in a Hilbert space. Theorem 4.1.2. Suppose W is a subspace of a Hilbert space V and v is a vector in V . Then, we have the following: 1. The vector w W is the best approximation of v V by vectors in W if and only if v w is orthogonal to every vector in W . 63 64 CHAPTER 4. REPRESENTATION AND APPROXIMATION 2. If a best approximation of v V by vectors in W exists, it is unique. 3. If W has a countable orthogonal basis w 1 ,w 2 ,... and is closed, then w = i =1 v  w i w i 2 w i (4.1) is the best approximation of v by vectors in W . Proof. Let w W and suppose v w is orthogonal to every vector in W . Let w W such that w = w . Then v w = ( v w ) + ( w w ) and v w 2 = v w 2 + 2Re v w  w w + w w 2 = v w 2 + w w 2 v w 2 . (4.2) Conversely, suppose that v w v w for every w W . From (4.2), we get 2Re v w  w w + w w 2 for all w W . Note that every vector in W can be expressed as w w where w W , it follows that 2Re v w  w + w 2 (4.3) for every w W . If w is in W and w = w then we may take w = w w  v w w w 2 ( w w ) . Inequality (4.3) then reduces to the statement 2  v w  w w  2 w w 2 +  v w  w w  2 w w 2 . This inequality holds if and only if v w  w w = 0 . Therefore, v w is orthogonal to every vector in W . Hence the vector w W is a best approximation of v V by vectors in W if and only if v w is orthogonal to every vector in W . Suppose w ,w W are best approximations of v by vectors in W . Then v w = v w and (4.2) implies that w w = 0 . That is, if a best approximation exists then it is unique. 4.1. BEST APPROXIMATION 65 Assume that W is closed and has a countable orthogonal basis w 1 ,...,w n and let w be defned by (4.1). Then v w is orthogonal to w j For j N , i.e., v w  w j = v  w j n i =1 v  w i w i 2 w i w j = v  w j v  w j w i 2 w j  w j = 0 . That is, v w is orthogonal to every vector in W and thereFore w is the best ap proximation oF v by vectors in W ....
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 Fall '08
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