Abstract
We investigate the geometry of the spectra (the supports of the Fourier transforms) of functions belonging to the Orlicz space 
and prove, in particular, that if 
, 
and 
, then for any point in the spectrum of 
there is a sequence of spectral points with non-zero components that converges to that point. It is shown that the behaviour of the sequence of Luxemburg norms of the derivatives of a function is completely characterized by its spectrum. A new method is suggested for deriving the Nikol'skii inequalities in the Luxemburg norm for functions with arbitrary spectra. The results are then applied to establish Paley-Wiener-Schwartz type theorems for cases that are not necessarily convex, and to study some questions in the theory of Sobolev-Orlicz spaces of infinite order that has been developed in recent years by Dubinskii and his students.