On THIS site, Alexandre used Fourier transform to solve the problem.
If we use Fourier transform, how to define it to ensure any entire function has a FT?
Classical FT is defined by
$$ \mathcal{F}[f] = F(\xi) = \frac{1}{\sqrt{2 \pi}}\int_{-\infty}^{+\infty}f(z)\mathrm{e}^{-\mathrm{i} \xi z} \mathrm{d} z. $$
This only work for $f \in L^1(\mathbb{R})$. (If improved, it can work for $f \in L^2(\mathbb{R})$.)
I know $\mathcal{F}[\mathrm{e}^{sz}] = \sqrt{2 \pi} \delta(\xi – \mathrm{i}s)$, but I'm not sure about a general definition.
Best Answer
There are two questions.
For the specific functional equation considered in this question On equation f(z+1)-f(z)=f'(z), the formula I gave covers all entire solutions. I added the references there.
On the general question about "Fourier transform" of entire functions or functions on the real line which are not in $L^p$. One usually replaces Fourier transform with various versions of Laplace transform. There are many versions, for various problems. I recommend Hormander, Analysis of differential operators..., Chap. 9, or the paper MR0199747 Ljubič, Ju. I.; Tkačenko, V. A. Theory and certain applications of the local Laplace transform. (Russian) Mat. Sb. (N.S.) 70 (112) 1966 416–437. There is an English translation in Math USSR Sbornik. There is also a nice little book by Carleman of Fourier transform (in French).
Edit. See also On linear independence of exponentials for an example how Laplace transform of entire functions is used.