On analytic solutions of linear homological equation

Abstract

We study solutions to the difference equation $f(x+\omega)-f(x)=g(x)$, where $x\in\mathbb{R}$ is a variable, $\omega\in (0,1)$ is a given number, and $g$ is a given analytic $1$-periodic function. It is well-known that in the case when the number $\omega$ is Diophantine (sufficiently poorly approximated by rational numbers), one can construct a $1$-periodic solution analytic in a neighbourhood of the real axis.
If $\omega$ is rational, then in the general case there are no periodic solutions. We have constructed a quasiperiodic analytic solution, which in the case of Diophantine $\omega$ decomposes into the sum of the known $1$-periodic solution and an analytic $\omega$-periodic function. One can say that in the case of numbers $\omega$ that are well-approximated by rational numbers, the Fourier series of each of these two terms diverge, but their sum remains analytic.