Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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

 
 

 

On Bernstein type theorems concerning
the growth of derivatives of entire functions


Author: Sen-Zhong Huang
Journal: Proc. Amer. Math. Soc. 125 (1997), 493-505
MSC (1991): Primary 30D20; Secondary 47D03, 47A10
DOI: https://doi.org/10.1090/S0002-9939-97-03883-5
MathSciNet review: 1396981
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A subspace $X$ of $L^1_{loc}(\mathbb {R} )$ which is invariant under all left translation operators $T_t,$ $t\in \mathbb {R},$ is called admissible if $X$ is a Banach space satisfying the following properties:

(i) If $\Vert f_n\Vert _X\to 0,$ then there exists a subsequence $(n_k)$ such that $f_{n_k}(s)\to 0$ almost everywhere.

(ii) The group ${\mathcal T}_X:=\{T_t\vert _X: t\in \mathbb {R}\}$ is a bounded strongly continuous group. In this case, let

\begin{displaymath}C_X:=\sup \{\Vert T_t\Vert _X: t\in \mathbb {R}\}.\end{displaymath}

Typical admissible spaces are $C_0(\mathbb {R} ),$ $BUC(\mathbb {R} )$ and all spaces $L^p(\mathbb {R} )$ for $1\leq p<\infty .$ More generally, all of the Peetre interpolation spaces of two admissible spaces $X_1,X_2$ are also admissible.

A function $g\in L^1_{loc}(\mathbb {R} )$ is called subexponential if for every $\delta >0,$ $e^{-\delta \vert t\vert } g(t) \break \in L^1(\mathbb {R} ).$ With these definitions our main result goes as follows:

Theorem.. If $g$ is an entire function of exponential type $\tau $ such that its restriction to the real axis, denoted by $g_\mathbb {R} $, is subexponential and belongs to some admissible space $X,$ then the derivative $g^\prime _\mathbb {R} $ is also in $X.$ Moreover,
$\Vert \alpha g_\mathbb {R} +g^\prime _\mathbb {R} \Vert_X\leq (\alpha ^2+\tau ^2)^{1/2} \cdot C_X\cdot \Vert g_\mathbb {R} \Vert_X$ for each real $\alpha .$

This result yields as consequences and in a systematic way many new and old Bernstein type inequalities.


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

  • [Ar] W. ARVESON: On groups of automorphisms of operator algebras. J. Funct. Anal. 15 (1974), 217-243. MR 50:1016
  • [B-C] F. F. BONSALL AND M. J. CRABB: The spectral radius of a Hermitian element of a Banach algebra. Bull. London Math. Soc. 2 (1970), 178-180. MR 42:853
  • [Be] S. BERNSTEIN: Sur une les propriété des fonctions entiéres. C. R. Acad. Sci. Paris 176 (1923), 1603-1605.
  • [B-L] J. BERGH AND J. LÖFSTRÖM: Interpolation Spaces, an Introduction. Springer-Verlag 1976. MR 58:2349
  • [Bo] R. H. BOAS: Entire Functions. Academic Press Inc., Publishers, New York 1954. MR 16:914f
  • [Do] H. R. DOWSON: Spectral Theory of Linear Operators. Academic Press, London, New York, San Francisco 1978. MR 80c:47022
  • [D-S] R. J. DUFFIN AND A. C. SCHAEFFER: Some inequalities concerning functions of exponential type. Bull. Amer. Math. Soc. 43 (1937), 554-556.
  • [Ev] D. E. EVANS: On the spectrum of a one-parameter strongly continuous representation. Math. Scand. 39 (1976), 80-82. MR 55:3873
  • [Hu] S.-Z. HUANG: Spectral Theory for Non-Quasianalytic Representations of Locally Compact Abelian Groups. monograph 1995.
  • [Jo] P. E. T. JORGENSEN: Spectral theory for one-parameter groups of isometries. J. Math. Anal. Appl. 168 (1992), 131-146. MR 93e:47048
  • [Ka] Y. KATZNELSON: An Introduction to Harmonic Analysis. John Wiley & Sons, Ney York 1968. MR 40:1734
  • [K-S-T] E. KOCHNEFF, Y. SAGHER AND R. TAN: On Bernstein inequality. Illinois J. Math. 36 (1992), 297-309. MR 93h:42008
  • [Le] B. JA. LEVIN: Distributions of Zeros of Entire Functions. Translations of Mathematical Monographs, Vol. 5, Amer. Math. Soc., Providence (R.I.), 1964.
  • [Lo] L. H. LOOMIS: An Introduction to Abstract Harmonic Analysis. D. Van Nostrand Comp., Princeton (N.J.) 1953. MR 14:883c
  • [Lor] G. G. LORENTZ: Approximation of Functions. Holt, Rinehart and Winston, New York 1966. MR 35:4642
  • [L-P] J. L. LIONS AND J. PEETRE: Sur une classe d'espaces d'interpolation. Publ. Math. Inst. Hautes Etudes Sci. 19 (1964), 5-68. MR 29:2627
  • [L-T] J. LINDENSTRAUSS AND L. TZAFRIRI: Classical Banach Spaces II, Function Spaces. Springer-Verlag 1979. MR 81c:46001
  • [N-H] R. NAGEL AND S.-Z. HUANG: Spectral mapping theorem for $C_0-$groups satisfying non-quasianalytic growth conditions. Math. Nachr. 169 (1994), 207-218. MR 95h:47055
  • [Ol] D. OLESEN: On norm continuity and compactness of spectrum. Math. Scand. 35 (1974), 223-236. MR 51:10532
  • [Pal] T. W. PALMER: Banach Algebras and the General Theory of $^*-$Algebras I: Algebras and Banach Algebras. Cambridge Univ. Press, London 1994. MR 95c:46002
  • [Paz] A. PAZY: Semigroups of Linear Operators and its Applications to Partial Differential Equations. Springer-Verlag 1983. MR 85g:47061
  • [Ped] G. K. PEDERSEN: $C^*-$Algebras and their Automorphism Groups. Academic Press, London 1979. MR 81e:46037
  • [Pee] J. PEETRE: A theory of interpolation of normed spaces. Notas de mathematica Brazil 39 (1968), 1-86. MR 39:4662
  • [R-S] F. RÄBIGER AND R. SCHNAUBELT: The spectral mapping theorem for evolution semigroups on space of vector-valued functions. Semigroup Forum 52 (1996), 225-239. CMP 96:07
  • [Re] R. M. REDHEFFER: Completeness of sets of complex exponentials. Adv. In Math. 24 (1977), 1-62. MR 56:5852
  • [Si] A. M. SINCLAIR: The norm of a hermitian element in a Banach algebra. Proc. Amer. Math. Soc. 28 (1971), 446-450. MR 43:921
  • [Ti] A. F. TIMAN: Theory of Approximation of Functions of a Real Variable. International Series of Monographs on Pure and Applied Mathematics, Vol. 34, Pergamon Press, 1963. MR 33:465

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 30D20, 47D03, 47A10

Retrieve articles in all journals with MSC (1991): 30D20, 47D03, 47A10


Additional Information

Sen-Zhong Huang
Affiliation: Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, F. R. Germany
Email: huse@michelangelo.mathematik.uni-tuebinegn.de

DOI: https://doi.org/10.1090/S0002-9939-97-03883-5
Keywords: Entire function, Bernstein's inequality, strongly continuous group, spectrum
Received by editor(s): August 16, 1995
Additional Notes: Supported by a fellowship of the Deutscher Akademisher Austauschdienst (DAAD)
Communicated by: Theodore W. Gamelin
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society