Musielak-Orlicz spaces provide a generalization of the usual $L^p$ spaces on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ to spaces of functions for which the Luxemburg norm $$ \|f\|_M:=\inf\left\{\lambda >0: \int_{\mathbb{R}} M(x,|g(x)|/\lambda) dx \leq 1\right\}, $$ is finite. Here, $M$ is a so-called "$\phi$-function" which is measurable, $M(x,0)=0$, $M(x,s)>0$, $M$ is convex in its second argument, and it satisfies the growth conditions $\lim_{s\mapsto 0^+} M(x,s)s^{-1} = 0$ and $\lim_{s\to \infty} M(x,s) = \infty$.
The Wiener-Tauberian Theorems ($L^p$, for $p=1,2$) give a characterization of the cyclic vectors of the translation operator in $L^p(\mathbb{R})$. Are there known generalizations giving sufficient conditions for a vector in $x\in L_M$ to be cyclic?
