Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Functions with bounded spectrum

Author: Ha Huy Bang
Journal: Trans. Amer. Math. Soc. 347 (1995), 1067-1080
MSC: Primary 42B10; Secondary 26D20, 46E35
MathSciNet review: 1283539
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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 $ \vert\vert{D^\alpha }f\vert\vert _p^{1/\vert\alpha \vert},\,\alpha \geqslant 0$, has the same behavior as the sequence $ \mathop {\lim }\limits_{\xi \in \operatorname{supp} Ff} \vert{\xi ^\alpha }{\vert^{1/\vert\alpha \vert}}$, $ \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 $ \vert\vert{D^\alpha }f\vert{\vert _p},\,\alpha \geqslant 0$. A Paley-Wiener-Schwartz theorem for a nonconvex case, which is a consequence of the result, is given.

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Keywords: Inequalities for derivatives, Fourier transform
Article copyright: © Copyright 1995 American Mathematical Society