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Functions with bounded spectrum. (English) Zbl 0828.42009
Summary: Let $$0< p\leq \infty$$, $$f(x)\in L_p(\mathbb{R}^n)$$, and $$\text{supp } Ff$$ be bounded, where $$F$$ is the Fourier transform. We prove in this paper that the sequence $$|D^\alpha f|^{1/|\alpha|}_p$$, $$\alpha\geq 0$$, has the same behavior as the sequence $$\sup_{\xi\in \text{supp }Ff} |\xi^{\alpha}|^{1/|\alpha|}$$, $$\alpha\geq 0$$. In other words, if we know all “far points” of $$\text{supp } Ff$$, we can wholly describe this behavior without any concrete calculation of $$|D^\alpha f|_p$$, $$\alpha\geq 0$$. A Paley-Wiener-Schwartz theorem for a nonconvex case, which is a consequence of the result, is given.

##### MSC:
 42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 26D10 Inequalities involving derivatives and differential and integral operators 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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