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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A property of infinitely differentiable functions

Author: Ha Huy Bang
Journal: Proc. Amer. Math. Soc. 108 (1990), 73-76
MSC: Primary 26E10
MathSciNet review: 1024259
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Abstract: The existence of $ {\lim _{n \to \infty }}\vert\vert{f^{(n)}}\vert\vert _p^{1/n}$ for an arbitrary function $ f(x) \in {C^\infty }({\mathbf{R}})$ such that $ {f^{\left( n \right)}}(x) \in {L^p}({\mathbf{R}}),n = 0,1, \ldots (1 \leq p \leq \infty )$ and the concrete calculation of $ {\lim _{n \to \infty }}\vert\vert{f^{(n)}}\vert\vert _p^{1/n}$ are shown.

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

  • [1] A. Kolmogoroff, On inequalities between the upper bounds of the successive derivatives of an arbitrary function on an infinite interval, Amer. Math. Soc. Translation 1949 (1949), no. 4, 19. MR 0031009
  • [2] E. M. Stein, Functions of exponential type, Ann. of Math. (2) 65 (1957), 582–592. MR 0085342
  • [3] S. M. Nikol′skiĭ, Priblizhenie funktsii mnogikh peremennykh i teoremy vlozheniya, “Nauka”, Moscow, 1977 (Russian). Second edition, revised and supplemented. MR 506247

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Article copyright: © Copyright 1990 American Mathematical Society