2 if we want to increase the degree of the

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Chapter 4 / Exercise 7
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2. If we want to increase the degree of the approximating polynomial we need to start over again and solve a larger set of normal equations. That is, we cannot use the a 0 , a 1 , . . . , a n we already found. It is more efficient and easier to solve the Least Squares Approximation problem using orthogonality, as we did with approximation by trigonometric polynomials. Suppose that we have a set of polynomials defined on an interval [ a, b ], { φ 0 , φ 1 , ..., φ n } , such that φ j is a polynomial of degree j . Then, we can write any polyno- mial of degree at most n as a linear combination of these polynomials. In particular, the Least Square Approximating polynomial p n can be written as p n ( x ) = a 0 φ 0 ( x ) + a 1 φ 1 ( x ) + ... + a n φ n ( x ) = n X j =0 a j φ j ( x ) ,
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84 CHAPTER 5. LEAST SQUARES APPROXIMATION for some coefficients a 0 , . . . , a n to be determined. Then E ( a 0 , ..., a n ) = Z b a [ f ( x ) - n X j =0 a j φ j ( x )] 2 dx = Z b a f 2 ( x ) dx - 2 n X j =0 a j Z b a φ j ( x ) f ( x ) dx + n X j =0 n X k =0 a j a k Z b a φ j ( x ) φ k ( x ) dx (5.13) and 0 = ∂E ∂a m = - 2 Z b a φ m ( x ) f ( x ) dx + 2 n X j =0 a j Z b a φ j ( x ) φ m ( x ) dx. for m = 0 , 1 , . . . , n , which gives the normal equations n X j =0 a j Z b a φ j ( x ) φ m ( x ) dx = Z b a φ m ( x ) f ( x ) dx, m = 0 , 1 , ..., n. (5.14) Now if the set of approximating functions { φ 0 , .... φ n } is orthogonal , i.e. Z b a φ j ( x ) φ m ( x ) dx = 0 if j 6 = m (5.15) then the coefficients of the least squares approximation are explicitly given by a m = 1 α m Z b a φ m ( x ) f ( x ) dx, α m = Z b a φ 2 m ( x ) dx, m = 0 , 1 , ..., n. (5.16) and p n ( x ) = a 0 φ 0 ( x ) + a 1 φ 1 ( x ) + ... + a n φ n ( x ) . Note that if the set { φ 0 , .... φ n } is orthogonal, (5.16) and (5.13) imply the Bessel inequality n X j =0 α j a 2 j Z b a f 2 ( x ) dx. (5.17)
5.2. LINEAR INDEPENDENCE AND GRAM-SCHMIDT ORTHOGONALIZATION 85 This inequality shows that if f is square integrable, i.e. if Z b a f 2 ( x ) dx < , then the series X j =0 α j a 2 j converges. We can consider the Least Squares approximation for a class of linear combinations of orthogonal functions { φ 0 , ..., φ n } not necessarily polynomials. We saw an example of this with Fourier approximations 1 . It is convenient to define a weighted L 2 norm associated with the Least Squares problem k f k w, 2 = Z b a f 2 ( x ) w ( x ) dx 1 2 , (5.18) where w ( x ) 0 for all x ( a, b ) 2 . A corresponding inner product is defined by h f, g i = Z b a f ( x ) g ( x ) w ( x ) dx. (5.19) Definition 7. A set of functions { φ 0 , ..., φ n } is orthogonal, with respect to the weighted inner product (5.19), if h φ j , φ k i = 0 for j 6 = k . 5.2 Linear Independence and Gram-Schmidt Orthogonalization Definition 8. A set of functions { φ 0 ( x ) , ..., φ n ( x ) } defined on an interval [ a, b ] is said to be linearly independent if a 0 φ 0 ( x ) + a 1 φ 1 ( x ) + . . . a n φ n ( x ) = 0 , for all x [ a, b ] (5.20) then a 0 = a 1 = . . . = a n = 0 . Otherwise, it is said to be linearly dependent. 1 For complex-valued functions orthogonality means Z b a φ j ( x ) ¯ φ m ( x ) dx = 0 if j 6 = m , where the bar denotes the complex conjugate 2 More precisely, we will assume w 0, Z b a w ( x ) dx > 0, and Z b a x k w ( x ) dx < + for k = 0 , 1 , . . . . We call such a w an admissible weight function.
86 CHAPTER 5. LEAST SQUARES APPROXIMATION Example 12. The set of functions { φ 0 ( x ) , ..., φ n ( x ) } , where φ j ( x ) is a poly- nomial of degree j for j = 0 , 1 , . . . , n is linearly independent on any interval

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