2

As an exercise we have to calculate the fourier series of |sin(x)| (was no problem) and after that we are meant to show that this series converges uniformly towards |sin(x)|.

After thinking about it for a while and not coming up with anything productive except that it converges for x=0 and that the series converges uniformly towards something at all, I dont have any more ideas.

Sadly we can't use stuff like the rule for uniform convergence when the function is continuosly differentiable because we didn't learn that yet.

Any hint or solution would be appreciated, cheers :)

Edit: Maybe I should mention that we use sines and cosines and not the complex exp-function for the series.

Sephaia
  • 41
  • 1
    What was your answer for the Fourier series coefficients? Can you apply the Weierstrass M test? –  May 16 '16 at 23:03
  • the bn are 0 because |sin(x)| is even, a0=2/pi. an for n>=1 is 4/pi * (1/(n^(2)-1)) *((-1)^(n+1)-1). (I used a period of 2pi)

    The Weierstrass results in uniform convergence, the problem is that I dont know how to prove that it converges against |sin(x)|.

     – Sephaia May 16 '16 at 23:09
  • OK, so $|a_n| \leq 8/(\pi(n^2 - 1))$ and therefore $\sum_{n=1}^{\infty}|a_n| < \infty$. Also, $|a_n \cos(nx)| \leq |a_n|$. What does the Weierstress M test tell you? –  May 16 '16 at 23:26
  • That the Fourier Series of |sin(x)| converges absolutely and uniformly against some function f.(Which is also continuos because cos(nx) is continuos for every n). That was where I got by myself so far too. – Sephaia May 16 '16 at 23:32
  • OK, so the Fourier series of $f$ converges absolutely and uniformly, hence in particular it converges pointwise to some function. Since $f$ is continuous, this implies that the Fourier series must converge to $f$ –  May 16 '16 at 23:34
  • Thanks for the help, I sadly can't use that as a solution in this case but the information is still nice to know :)(I'd have to prove this statement from only knowing the definition of a fourier series) – Sephaia May 16 '16 at 23:38
  • Are you able to use the result that if a continuous function has all of its Fourier coefficients equal to zero, then the function must be identically zero? –  May 16 '16 at 23:43
  • Sadly no, the only knowledge concerning fourier series we are allowed to use for this exercise is how one calculates the coefficients an and bn and that the coefficients of a uniformly converging trigonometric series have to have the form of an/bn. – Sephaia May 16 '16 at 23:45
  • @Sephaia a bit off-topic - I've just taken a look at your profile... do you also happen to study in Bochum? xD – Sora. May 27 '16 at 00:40
  • Yeah, I do study in Bochum :D – Sephaia May 31 '16 at 06:43

0 Answers0