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Unformatted text preview: 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 . 87 LECTURE NOTES FOR 18.102, SPRING 2009 Lecture 14. Tuesday, March 31: Fourier series and L 2 (0 , 2 ) . 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 ) then the Fourier series of u, (14.1) 1 c k e ikx , c k = u ( x ) e ikx dx 2 k Z (0 , 2 ) converges in L 2 (0 , 2 ) 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 c k e ikx  2 = 0 n  k  n for any u L 2 (0 , 2 ) . First lets 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 ) satisfy 2 if k = j (14.3) e k e j = exp( i ( k j ) x ) = 2 if k = j. Thus the functions e k 1 ikx (14.4) e k = = e e k 2 form an orthonormal set in L 2 (0 , 2 ) . It follows that (14.1) is just the FourierBessel series for u with respect to this orthonormal set: (14.5) c k = 2 u,e k = 1 c k e ikx = u,e k e k . 2 So, we alreay know that this series converges in L 2 (0 , 2 ) thanks to Bessels identity. So all we need to show is Proposition 21. The e k , k Z , form an orthonormal basis of L 2 (0 , 2 ) , i.e. are complete: (14.6) ue ikx = 0 k = u = 0 in L 2 (0 , 2 ) . 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 ) . I will prove this using Fej ers method. In this approach, we check that any continuous function on [0 , 2 ] satisfying the additional condition that u (0) = u (2 ) is the uniform limit on [0 , 2 ] 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 ) and we already know that the continuous functions which vanish near and 2 are dense in L 2 (0 , 2 ) (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...
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This note was uploaded on 11/08/2011 for the course PHY 18.102 taught by Professor Staff during the Spring '09 term at MIT.
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