I am sorry for not being rigorous in asking this question as I lack knowledge in several of the concerned areas.
Under some conditions, the exponential of the scaled derivative operator $D$ is the translation operator: denoting $D f = f'$, we have $exp(aD)f = f(. + a)$ for any $a \in \mathbb R$. Hence, a funciton of period $a$ verifies $\exp(aD)f = f(. + a) = f$. Now, if we treat it as a linear differential equation of infinite order (the exponential can be expanded as an infinite series), we would solve the characteristic equation $\exp(ar)=1$. The latter has solutions $2ik\pi/a; k \in \mathbb Z$. Hence, continuing the non rigorous reasoning, we see that $f$ is an (infinite) linear combination of $\exp(2ik\pi x/a)$ functions, in other words, a Fourier series. Were such links established in the literature?
For unbounded operators, the exponential as a series cannot be defined in general. This exponential notation refers to as the so-called semigroup associated to the unbounded operator.