Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-1995-1283539-1
MathSciNet review: 1283539
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] R. A. Adams, Sobolev spaces, Academic Press, New York, San Francisco, and London, 1975. MR 0450957 (56:9247)
  • [2] H. H. Bang, A property of infinitely differentiable functions, Proc. Amer. Math. Soc. 108 (1990), 73-76. MR 1024259 (90j:26029)
  • [3] -, Some imbedding theorems for the spaces of infinite order of periodic functions, Math. Notes 43 (1988), 509-517. MR 940848 (89f:46072)
  • [4] -, On imbedding theorems for Sobolev spaces of infinite order, Mat. Sb. 136 (1988), 115-127.
  • [5] -, Imbedding theorems for Sobolev spaces of infinite order, Acta Math. Vietnam 14 (1989), 17-29. MR 1058273 (91e:46040)
  • [6] Ju. B. Egorov, Lectures on partial differential equations, Moscow State Univ. Press., Moscow, 1975.
  • [7] L. Hörmander, The analysis of linear partial differential operators I, Grundlehren Math. Wiss., 256, Springer, Berlin, Heidelberg, New York, and Tokyo, 1983. MR 717035 (85g:35002a)
  • [8] P. I. Lizorkin, Estimates for trigonometric integrations and the Bernstein inequality for fractional derivatives, Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965), 109-126. MR 0178306 (31:2564)
  • [9] R. J. Nessel and G Wilmes, Nikolskii-type inequalities for trigonometric polynomials and entire functions of exponential type, J. Austral. Math. Soc. 25 (1978), 7-18. MR 0487212 (58:6872)
  • [10] S. M. Nikolsky, Approximation of functions of several variables and imbedding theorems, "Nauka", Moscow, 1977.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 42B10, 26D20, 46E35

Retrieve articles in all journals with MSC: 42B10, 26D20, 46E35


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1995-1283539-1
Keywords: Inequalities for derivatives, Fourier transform
Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society