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




Re: Series Converges to Function NonDifferentiable at 0.
Posted:
May 14, 2013 2:17 PM


On Tuesday, May 14, 2013 10:18:31 AM UTC7, 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.



