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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.

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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.

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    $\begingroup$ Cool ! But seems trivial solution f = z is not covered by the answer... $\endgroup$
    – Alexander Chervov
    Commented Nov 29, 2012 at 13:00
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    $\begingroup$ And yet the "trivial" solution $z$ is not obtained in this way. $\endgroup$
    – Gerald Edgar
    Commented Nov 29, 2012 at 13:03
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    $\begingroup$ If $\lambda$ is a double root of $e^\lambda-1-\lambda$, then we should include $z e^{\lambda z}$ as well. This recovers the trivial solution, since $0$ is a double root. $\endgroup$
    – Henry Cohn
    Commented Nov 29, 2012 at 13:27
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    $\begingroup$ I do not know a way to define FT for ALL entire functions. And I suppose there is no reasonable way. FT can be defined for functions on the real line which satisfy $|f(x)|=O(e^{\epsilon x})$ for every $\epsilon>0$ which is enough for most applications. Speaking of that functional equation, I did not claim that I described ALL entire solutions. But this was not asked:-) $\endgroup$
    – Alexandre Eremenko
    Commented Dec 2, 2012 at 15:12
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    $\begingroup$ We have $-(\lambda +1)e^{-(\lambda+1)}=-e^{-1}$, so the $\lambda$ are determined by the multiple branches of Lambert's $W$-function, namely: $\lambda = -1-W(-e^{-1})$. $\endgroup$
    – Thomas
    Commented Feb 17, 2017 at 23:38

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