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Topic: Elementary Fourier analysis
Replies: 2   Last Post: Apr 23, 2013 11:34 AM

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Paul

Posts: 463
Registered: 7/12/10
Re: Elementary Fourier analysis
Posted: Apr 23, 2013 11:34 AM
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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:
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>
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> >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).
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>
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> Usually, yes.
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>
>

> >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?
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>
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> Yes. Because the symmetric partial sum is equal to f *D_N, where *
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> denotes convolution and D_N is the Dirichlet kernel; this is the basis
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> of many results about convergence of Fourier series, and it
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> doesn't work for asymmetric partial sums.
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> See "Dini's test" and its proof, for example.
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>

> >Would Parseval's theorem fail to hold with this alternative definition?
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> No, Parseval holds for any sort of rearrangement of the Fourier
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> series.
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> Because Parseval is just a result about complete orthonormal
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> sets, and any reordering of the standard orthonomal basis
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> is still an orthonormal basis. The things that do change are
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> subtler than Parseval.
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>

> >
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> >Thank you,
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> >
>
> >Paul Epstein

Thanks a lot. Great response.

Paul



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