Drexel dragonThe Math ForumDonate to the Math Forum



Search All of the Math Forum:

Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.


Math Forum » Discussions » sci.math.* » sci.math.independent

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

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
RGVickson@shaw.ca

Posts: 1,653
Registered: 12/1/07
Re: Series Converges to Function Non-Differentiable at 0.
Posted: May 14, 2013 2:17 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

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

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



Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© Drexel University 1994-2014. All Rights Reserved.
The Math Forum is a research and educational enterprise of the Drexel University School of Education.