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Topic: Series Converges to Function Non-Differentiable at 0.
Replies: 4   Last Post: May 15, 2013 1:16 PM

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 RGVickson@shaw.ca Posts: 1,677 Registered: 12/1/07
Re: Series Converges to Function Non-Differentiable at 0.
Posted: May 14, 2013 2:17 PM

On Tuesday, May 14, 2013 10:18:31 AM UTC-7, hbe...@gmail.com wrote:
> Hi, All:
>

> >>
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> >> I want to find an example of a series Sum_n=1...oo A_nCos(nx)
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> >>
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> >> in L^2[-Pi,Pi] to a continuous function that is not differentiable at
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> >>
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> >> x=0.
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> >>
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> >> My idea: we can use the Fourier Series for |x|, which is in L^2[-Pi,Pi]
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> >>
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> >> is not differentiable at 0 , and then use the fact that the Fourier Series of
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> >>
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> >> a function f(x) in L^2 converges pointwise to itself. So {a_n} are
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> >>
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> >> the coefficients of the Fourier Series for |x| .
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> >>
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> >> Does that do it ?
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>
>
> I posted this in the 'Analyst' forum in 'Topology Atlas', and someone
>
>
>
> (correctly) replied that the Fourier series converges pointwise only a.e.
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>
>
> So, how do I change things to that the Fourier Series for |x| converges to
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>
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> 0 at 0?
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>
>
> Thanks.

The function f(x) = |x| on I = [-pi,pi] satisfies the Dirichlet conditions on I, so its Fourier series converges to f(x) for all x in the interior of I. This is much better that a.e. convergence.

Date Subject Author
5/14/13 gk@gmail.com
5/14/13 RGVickson@shaw.ca
5/15/13 Roland Franzius
5/15/13 David C. Ullrich
5/15/13 gk@gmail.com