Original Problem
If $f$ is an entire function such that
$$ f(z+1)-f(z)=f'(z) $$
for all $z$.
Is there a non-trivial solution? ($f(z)=az+b$ is trivial)
And here is something uncertainty
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
Linear functional equations can be solved with Fourier transform. Let $\lambda_k$ be the roots of the equation $e^\lambda-1=\lambda$. There are infinitely many such roots. Then
$$f(z)=\sum_k a_ke^{\lambda_k z}$$
is a solution. Here the sum can be finite, and $a_k$ arbitrary, or the sum can be infinite, and $a_k$ tend to zero with such speed that the series converges.
The trivial solution is covered, if you interpret the answer correctly. The equation $e^\lambda-1=\lambda$ has a DOUBLE root at $0$. For the case of a double root one includes not only the term $e^{\lambda z}$ but also the term $ze^{\lambda z}$. Similarly for multiple roots $z^ke^{\lambda z}$. So the root $\lambda=0$ exactly covers the solution $az+b$.
EDIT: Actually all entire solutions can be represented in this way, for the proof I refer to Gelfond, Calculus of finite differences, MR0342890, Chap. 5 Sect 7, Thm II and Corollaries. He gives the equation $f'(z)=f(z-1)$ as an example, but your equation is treated similarly.
EDIT2: The problem can be generalized as follows: Let $\omega$ be a distribution with bounded support. Equation $f\star w=0$, where $\star$ is the convolution, is called a convolution equation, and its solutions are called mean-periodic functions (fonctions moyenne-periodiques). The case we are discussing is $\omega(x)=\delta(x+1)-\delta(x)-\delta'(x)$ where $\delta$ is the delta function. The theory of mean periodic functions was created by Delsarte and Schwartz in 1940-s. Schwartz has a general result that all mean-periodic functions can be obtained as limit of exponential sums, like in this problem.