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MIT18_102s09_lec14

# MIT18_102s09_lec14 - MIT OpenCourseWare http/ocw.mit.edu...

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MIT OpenCourseWare http://ocw.mit.edu 18.102 Introduction to Functional Analysis Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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87 LECTURE NOTES FOR 18.102, SPRING 2009 Lecture 14. Tuesday, March 31: Fourier series and L 2 (0 , 2 π ) . Fourier series. Let us now try applying our knowledge of Hilbert space to a concrete Hilbert space such as L 2 ( a, b ) for a finite interval ( a, b ) R . You showed that this is indeed a Hilbert space. One of the reasons for developing Hilbert space techniques originally was precisely the following result. Theorem 12. If u L 2 (0 , 2 π ) then the Fourier series of u, (14.1) 1 c k e ikx , c k = u ( x ) e ikx dx 2 π k Z (0 , 2 π ) converges in L 2 (0 , 2 π ) to u. Notice that this does not say the series converges pointwise, or pointwise almost everywhere since this need not be true depending on u. We are just claiming that 1 (14.2) lim | u ( x ) 2 π c k e ikx | 2 = 0 n →∞ | k |≤ n for any u L 2 (0 , 2 π ) . First let’s see that our abstract Hilbert space theory has put us quite close to proving this. First observe that if e k ( x ) = exp( ikx ) then these elements of L 2 (0 , 2 π ) satisfy 2 π 0 if k = j (14.3) e k e j = exp( i ( k j ) x ) = 0 2 π if k = j. Thus the functions e k 1 ikx (14.4) e k = = e e k 2 π form an orthonormal set in L 2 (0 , 2 π ) . It follows that (14.1) is just the Fourier-Bessel series for u with respect to this orthonormal set:- (14.5) c k = 2 π u, e k = 1 c k e ikx = u, e k e k . 2 π So, we alreay know that this series converges in L 2 (0 , 2 π ) thanks to Bessel’s identity. So ‘all’ we need to show is Proposition 21. The e k , k Z , form an orthonormal basis of L 2 (0 , 2 π ) , i.e. are complete: (14.6) ue ikx = 0 k = u = 0 in L 2 (0 , 2 π ) . This however, is not so trivial to prove. An equivalent statement is that the finite linear span of the e k is dense in L 2 (0 , 2 π ) . I will prove this using Fej´ er’s method. In this approach, we check that any continuous function on [0 , 2 π ] satisfying the additional condition that u (0) = u (2 π ) is the uniform limit on [0 , 2 π ] of a sequence in the finite span of the e k . Since uniform convergence of continuous functions certainly implies convergence in L 2 (0 , 2 π ) and we already know that the continuous functions which vanish near 0 and 2 π are dense in L 2 (0 , 2 π ) (I will recall why later) this is enough to prove Proposition 21. However the proof is a serious piece of analysis, at least it is to me!
88 LECTURE NOTES FOR 18.102, SPRING 2009 So, the problem is to find the sequence in the span of the e k . Of course the trick is to use the Fourier expansion that we want to check. The idea of Ces` aro is to make this Fourier expansion ‘converge faster’, or maybe better. For the moment we can work with a general function u L 2 (0 , 2 π ) or think of it as continuous if you prefer. So the truncated Fourier series is 1 u ( t ) e ikt dt ) e ikx (14.7) U n ( x ) = ( 2 π

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MIT18_102s09_lec14 - MIT OpenCourseWare http/ocw.mit.edu...

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