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Paul
Posts:
715
Registered:
7/12/10


Re: Elementary Fourier analysis
Posted:
Apr 23, 2013 11:34 AM


On Tuesday, April 23, 2013 2:42:29 PM UTC+1, dull...@sprynet.com wrote: > On Tue, 23 Apr 2013 03:27:14 0700 (PDT), pepstein5@gmail.com wrote: > > > > >As I understand it, the sum from n = infinity to n = infinity of c_n exp(inx) is defined as the limit as N tends to infinity of the sum from N to N of c_n exp(inx). > > > > Usually, yes. > > > > >Suppose the infinite sum is defined as lim N > infinity, M> infinity [ the sum from n =  M to n = N] of c_n exp(inx), then does the basic theory change? > > > > Yes. Because the symmetric partial sum is equal to f *D_N, where * > > denotes convolution and D_N is the Dirichlet kernel; this is the basis > > of many results about convergence of Fourier series, and it > > doesn't work for asymmetric partial sums. > > > > See "Dini's test" and its proof, for example. > > > > >Would Parseval's theorem fail to hold with this alternative definition? > > > > No, Parseval holds for any sort of rearrangement of the Fourier > > series. > > > > Because Parseval is just a result about complete orthonormal > > sets, and any reordering of the standard orthonomal basis > > is still an orthonormal basis. The things that do change are > > subtler than Parseval. > > > > > > > >Thank you, > > > > > >Paul Epstein
Thanks a lot. Great response.
Paul



