Properties of functions in Orlicz spaces that depend on the geometry of their spectra

We investigate the geometry of the spectra (the supports of the Fourier transforms) of functions belonging to the Orlicz space $L_{\Phi}(\mathbb R^n)$ and prove, in particular, that if $f\in L_p(\mathbb R^n)$, $1\leqslant p<\infty$ and $f(x)\not\equiv 0$, then for any point in the spectrum of $f$ 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.