Laplace Transform of a Fourier Series

Question

If I have a Fourier series, how can its Laplace transform be determined?

Is it correct to determine the Laplace transform of each single term, or should I proceed in some other way?

Answer 1

The Laplace transform is a linear integral operator, so the identity $$ \left(\mathcal{L}\sum_{n\geq 1} f_n(x)\right)(s) = \sum_{n\geq 1}\left(\mathcal{L} f_n\right)(s) $$ holds if the hypothesis of the dominated convergence theorem are met. If $f_n(x)=a_n \sin(nx)$ then $\left(\mathcal{L} f_n\right)(s)=a_n\cdot\frac{n}{n^2+s^2}$ is bounded by $\min\left(\frac{|a_n|}{n},\frac{|a_n|}{2s}\right)$ in absolute value for any $s>0$. In particular $\{a_n\}_{n\geq 1}\in \ell^1\cup \ell^2$ is sufficient to ensure that the above identity holds.

For instance, we are allowed to state $$\mathcal{L}(|\sin x|)(s) = \frac{2}{\pi s}-\frac{4}{\pi}\sum_{n\geq 1}\frac{s}{(4n^2-1)(4n^2+s^2)}=\frac{\coth\frac{\pi s}{2}}{1+s^2}. $$