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?


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