# Solution to an equation involving integrals

I basically have a integral equation which I then want to put on a lattice. I want to find a continuous and periodic function $$F(q^1,q^2):\mathbb{R}^2\rightarrow\mathbb{R}$$ which solves $$\int_0^1F(q^1+s,q^2)ds=\int_0^1F(q^1-s,q^2)ds.$$ Then make the substitution $$q^1=\frac{\ell_qk^1}{N}$$ and $$q^2=\frac{\ell_qk^2}{N}$$ where $$k^1=0,1,...,N-1$$ and $$k^2=0,1,...,N-1$$, to then have the system on a lattice. Below I have also written it in a lattice form.

From above I need a function $$F(\frac{\ell_qk^1}{N},\frac{\ell_qk^2}{N}):\mathbb{Z}^2\mapsto\mathbb{R}$$ for $$N\in\mathbb{N}$$ but bigger than one, ideally equal to about 10, such that $$F(\frac{\ell_q(k^1+N)}{N},\frac{\ell_q(k^2+N)}{N})=F(\frac{\ell_qk^1}{N},\frac{\ell_qk^2}{N})$$ is a periodic function which solves the following equation: $$\int_0^1F(\frac{\ell_q(k^2+s)}{N},\frac{\ell_qk^2}{N})ds=\int_0^1F(\frac{\ell_q(k^2-s)}{N},\frac{\ell_qk^2}{N})ds.$$

The function cannot just depend on $$k^2$$ or $$s$$. I've tried representing the function as a Fourier series i.e., $$F(\frac{\ell_qk^1}{N},\frac{\ell_qk^2}{N})=\sum_{\boldsymbol{m}}c_{\boldsymbol{m}}\mathrm{e}^{\frac{2\pi i(m_1k^1+m_2k^2)}{N}},$$ Then tried finding a condition for the coefficients, but all I got was a condition involving $$n_1$$ and $$n_2$$ and $$N$$ of the form $$\sin(\pi (m_1+m_2)/N)=0$$ or $$m_1+m_2=2nN,\text{ where }n\in\mathbb{Z}.$$ But I don't see how this helps me find an explicit function which solve the integral equation.

Does anybody know a function which would work? Or a class of functions which would satisfy this?