I would like to describe the class of all functions $a\in L^1(\mathbb{R},dx)$, such that there exists $\tilde{a}=a$ a.s. and a size $h$ of an infinite partition of $\mathbb{R}$, such that $\sum\limits_{k\in\mathrm{Z}} \sup\limits_{[kh,kh+h]} |\tilde{a}(x)| <\infty$ ?
For example, if $a(x)=\sum_{n\in\mathbb{N}} n \, 1\!\!1_{\bigl[n,n+\frac{1}{n^3}\bigr]}(x)$ then such property does not hold.