Functions with bounded spectrum

Abstract:

Let $0 < p \leqslant \infty , f(x) \in {L_p}({\mathbb {R}^n})$, and $\operatorname {supp} Ff$ be bounded, where $F$ is the Fourier transform. We will prove in this paper that the sequence $||{D^\alpha }f||_p^{1/|\alpha |}, \alpha \geqslant 0$, has the same behavior as the sequence $\lim \limits _{\xi \in \operatorname {supp} Ff} |{\xi ^\alpha }{|^{1/|\alpha |}}$, $\alpha \geqslant 0$. In other words, if we know all "far points" of $\operatorname {supp} Ff$, we can wholly describe this behavior without any concrete calculation of $||{D^\alpha }f|{|_p}, \alpha \geqslant 0$. A Paley-Wiener-Schwartz theorem for a nonconvex case, which is a consequence of the result, is given.