Approximation_of_Periodic_Functions.pdf - Chapter 4 Approximation of Periodic Functions We now consider the problem of approximating a periodic function

Approximation_of_Periodic_Functions.pdf - Chapter 4...

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Chapter 4 Approximation of Periodic Functions We now consider the problem of approximating a periodic function f . With- out loss of generality we can assume that f is of period 2 (if it is of period p then the function F ( y ) = f ( p 2 y ) has period 2 ). We are going to assume that f is piece-wise continuous with finite jumps, i.e. the right and left limits lim h ! 0 + f ( x + h ) = f + ( x ) , (4.1) lim h ! 0 + f ( x - h ) = f - ( x ) (4.2) exist and are finite. We would like to approximate such a periodic function f by a Trigonono- metric polynomial: S n ( x ) = 1 2 a 0 + n X k =1 ( a k cos kx + b k sin kx ) . (4.3) We can also write this in complex form, by noting e ikx = cos kx + i sin kx , we have S n ( x ) = n X k = - n c k e ikx (4.4) 55
56 CHAPTER 4. APPROXIMATION OF PERIODIC FUNCTIONS where c 0 = 1 2 a 0 (4.5) c k = 1 2 ( a k - ib k ) , k = ± 1 , . . . , ± n (4.6) c - k = 1 2 ( a k + ib k ) , k = ± 1 , . . . , ± n. (4.7) We are interested in the best approximation of f in the 2 norm. That is, we would like to approximate f by S n in the Least Squares sense k f - S n k 2 = ✓Z 2 0 [ f ( x ) - S n ( x )] 2 dx 1 2 = min (4.8) Let J n = k f - S n k 2 2 . Then we need to find the coe ffi cients c k , k = 0 , ± 1 , ± 2 , . . . ± n (or equivalently a 0 , a 1 , ..., a n , b 1 , ..., b n ) which minimize J n . We have J n = Z 2 0 " f ( x ) - n X k = - n c k e ikx # 2 dx = Z 2 0 [ f ( x )] 2 dx - 2 n X k = - n c k Z 2 0 f ( x ) e ikx dx + n X k = - n n X l = - n c k c l Z 2 0 e ikx e ilx dx. (4.9) This problem simplifies if we use the orthogonality of the set { 1 , e ix , e - ix , . . . , e inx , e - inx } or equivalently of the set of trigonometric functions { 1 , cos x, cos 2 x, . . . , cos nx, sin x, sin 2 x, . . . , sin nx } . As for k 6 = - l Z 2 0 e ikx e ilx dx = Z 2 0 e i ( k + l ) x dx = 1 i ( k + l ) e i ( k + l ) x 2 0 = 0 (4.10) and for k = - l Z 2 0 e ikx e ilx dx = Z 2 0 dx = 2 (4.11)
57 Then we get J n = Z 2 0 [ f ( x )] 2 dx - 2 n X k = - n c k Z 2 0 f ( x ) e ikx dx + 2 n X k = - n c k c - k . (4.12) J n is a quadratic function of the coe ffi cients c k and so to find the its min- imum, we determine the critical point of J n as a function of the c k , k = 0 , ± 1 , . . . , ± , n . @ J n @ c 0 = - 2 Z 2 0 f ( x ) dx + 2(2 ) c 0 = 0 (4.13) @ J n @ c l = - 2 Z 2 0 f ( x ) e ilx dx + 2(2 ) c - l = 0 , l = ± 1 , . . . , ± n. (4.14) Therefore, relabeling the coe ffi cients with k again, we get c k = 1 2 Z 2 0 f ( x ) e - ikx dx, k = 0 , ± 1 , . . . , ± n, (4.15) which are the complex Fourier coe ffi cients of f . We can now obtain the real Fourier coe ffi cients a k and b k by noting that a 0 = 2 c 0 (4.16) a k = c k + c - k , k = 1 , . . . , n (4.17) b k = i ( c k - c - k ) , k = 1 , . . . , n. (4.18) Hence a k = 1 Z 2 0 f ( x ) cos kxdx, k = 0 , 1 , . . . , n, (4.19) b k = 1 Z 2 0 f ( x ) sin kxdx, k = 1 , . . . , n. (4.20) Now, if we substitute the Fourier coe ffi cients (4.15) in (4.12) we get 0 J n = Z 2 0 [ f ( x )] 2 dx - 2 n X k = - n | c k | 2
58 CHAPTER 4. APPROXIMATION OF PERIODIC FUNCTIONS that is n X k = - n | c k | 2 1 2 Z 2 0 [ f ( x )] 2 dx. (4.21) This is known as Bessel’s inequality. If we substitute (4.5)-(4.7) we obtain Bessel’s inequality for the real Fourier coe ffi cients 1 2 a 2 0 + n X k =1 ( a 2 k + b 2 k ) 1 Z 2 0 [ f ( x )] 2 dx. (4.22) If R 2 0 [ f ( x )] 2 dx is finite, as is the case for the f under consideration, then then series 1 2 a 2 0 + 1 X k =1 ( a 2 k + b 2 k ) converges and consequently lim k !1 a k = lim k !1 b k = 0.

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