A property of infinitely differentiable functions
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- by Ha Huy Bang PDF
- Proc. Amer. Math. Soc. 108 (1990), 73-76 Request permission
Abstract:
The existence of ${\lim _{n \to \infty }}||{f^{(n)}}||_p^{1/n}$ 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 ${\lim _{n \to \infty }}||{f^{(n)}}||_p^{1/n}$ are shown.References
- 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
- E. M. Stein, Functions of exponential type, Ann. of Math. (2) 65 (1957), 582–592. MR 85342, DOI 10.2307/1970066
- S. M. Nikol′skiĭ, Priblizhenie funktsiĭ mnogikh peremennykh i teoremy vlozheniya, “Nauka”, Moscow, 1977 (Russian). Second edition, revised and supplemented. MR 506247
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 108 (1990), 73-76
- MSC: Primary 26E10
- DOI: https://doi.org/10.1090/S0002-9939-1990-1024259-9
- MathSciNet review: 1024259