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I'm trying to solve the following initial value problem (from Folland, pg. 277, exercise $48$a) using Fourier series:

Let $x \mapsto f(x)$ and $x \mapsto u(x, t)$ be periodic functions on $\mathbb{R}$. No regularity is given, so we can just assume whatever is needed as we go. The equation is: $$(\partial_x^2 + \partial_t^2)u = 0.$$ With an initial value: $$u(x, 0) = f(x).$$

What I got after applying the Fourier transform to both sides and solving the corresponding ODE is the following: $$u(x, t) = \sum_{\xi \in \mathbb{Z}} c_1(\xi)e^{2\pi\xi[t + ix]} + c_2(\xi)e^{2\pi\xi[-t + ix]},$$ where $\hat{f}(\xi) = c_1(\xi) + c_2(\xi)$. So far, I think this is ok.

My questions are:

  1. Is it possible to go further? Like, is there anything else I can do to make the solution more explicit, such as writing it as a convolution or something?

  2. There is a comment in this exercise about Abel means/Abel summation. What does that have to do with anything here?

Thank you!

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You will get the general solution of the form $u(x,t)=C_1(x+it)+C_2(x-it)$ , for matching the only condition $u(x,0)=f(x)$ , you can obviously take $C_1(x)=\dfrac{f(x)}{2}+F(x)$ and $C_2(x)=\dfrac{f(x)}{2}-F(x)$ so that you can get $u(x,t)=\dfrac{f(x+it)+f(x-it)}{2}+F(x+it)-F(x-it)$

Note that this solution satisfy on $x,t\in\mathbb{C}$ .

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  • $\begingroup$ Thank you, but may I ask how did you get that solution from what I've done? And also, do you have any thoughts on the Abel summation thing? $\endgroup$
    – L..
    Aug 19 '13 at 21:16

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