Functions with bounded spectrum
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- by Ha Huy Bang PDF
- Trans. Amer. Math. Soc. 347 (1995), 1067-1080 Request permission
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.References
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Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 347 (1995), 1067-1080
- MSC: Primary 42B10; Secondary 26D20, 46E35
- DOI: https://doi.org/10.1090/S0002-9947-1995-1283539-1
- MathSciNet review: 1283539