
> 
> Find all continuous and bounded functions $g$
> with :
> $$\forall x \in \mathbb R, 4g(x)=g(x+1)+g(x-1)+g(x+\pi)+g(x-\pi).$$
> 
> 
> 


I have posted this question [here](https://math.stackexchange.com/questions/4630194/x-ens-functional-equation), but received no answer.



How may I find all continuous and bounded functions g with the following property?
$\newcommand\de\delta$Considering $g$ a distribution (in the generalized-function sense), let $\hat g$ be the Fourier transform of $g$. Then your functional equation yields
$$4\hat g(t)=e^{it}\hat g(t)+e^{-it}\hat g(t)+e^{i\pi t}\hat g(t)+e^{-i\pi t}\hat g(t),$$
or
$$(\cos t+\cos\pi t-2)\hat g(t)=0,$$
for real $t$.


The equality $\cos t+\cos\pi t-2=0$ for real $t$ implies $\cos t=1=\cos\pi t$ and hence $t=0$ (because $\pi$ is irrational). So,
the support of $\hat g$ is $\{0\}$. So (see e.g. "For every compact subset $K\subseteq U$ there exist constants
$C\_{K}>0$ and $N\_{K}\in \mathbb {N}$ such that for all $f\in C\_{c}^{\infty }(U)$ with support contained in $K$ [...]" [here](https://en.wikipedia.org/wiki/Distribution_(mathematics)#Definitions_of_test_functions_and_distributions)), we have $\hat g=\sum\_{j=0}^n c\_j\de^{(j)}$ for some $n\in\{0,1,\dots\}$ and some complex $c\_j$'s, where $\de^{(j)}$ is the $j$th derivative of the delta function. So, $g$ is a polynomial. Since $g$ is bounded, it is constant. $\quad\Box$


