  For each integrability parameter $p \in (0,\infty]$, the critical smoothness
of a periodic generalized function $f$, denoted by $s_f(p)$ is the supremum
over the smoothness parameters $s$ for which $f$ belongs to the Besov space
$B_{p,p}^s$ (or other similar function spaces). This paper investigates the
evolution of the critical smoothness with respect to the integrability
parameter $p$. Our main result is a simple characterization of all the possible
critical smoothness functions $p\mapsto s_f(p)$ when $f$ describes the space of
generalized periodic functions. We moreover characterize the compressibility of
generalized periodic functions in wavelet bases from the knowledge of their
critical smoothness function.
