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)

 
 

 

Discrete sufficient sets for some spaces of entire functions


Author: B. A. Taylor
Journal: Trans. Amer. Math. Soc. 163 (1972), 207-214
MSC: Primary 46.30; Secondary 30.00
DOI: https://doi.org/10.1090/S0002-9947-1972-0290084-3
MathSciNet review: 0290084
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ E$ denote the space of all entire functions $ f$ of exponential type (i.e. $ \vert f(z)\vert = O(\exp (B\vert z\vert))$) for some $ B > 0$). Let $ \mathcal{K}$ denote the space of all positive continuous functions $ k$ on the complex plane $ C$ with $ \exp (B\vert z\vert) = O(k(z))$ for each $ B > 0$. For $ k \in \mathcal{K}$ and $ S \subset C$, let $ \vert\vert f\vert{\vert _{k,s}} = \sup \{ \vert f(z)\vert/k(z):z \in S\}$. We prove that the two families of seminorms $ {\{ \vert\vert\vert{\vert _{k,C}}\} _{k \in \mathcal{K}}}$ and $ {\{ \vert\vert\vert{\vert _{k,s}}\} _{k \in \mathcal{K}}}$, where

$\displaystyle S = \{ n + im: - \infty < n,m < + \infty \} $

, determine the same topology on $ E$.

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

  • [1] R. P. Boas, Entire functions, Academic Press, New York, 1954. MR 16, 914. MR 0068627 (16:914f)
  • [2] M. L. Cartwright, On functions bounded at the lattice points in an angle, Proc. London Math. Soc. 43 (1937), 26-32.
  • [3] L. Ehrenpreis, Fourier analysis in several complex variables, Interscience, New York, 1970. MR 0285849 (44:3066)
  • [4] -, Analytically uniform spaces and some applications, Trans. Amer. Math. Soc. 101 (1961), 52-74. MR 24 #A1604. MR 0131756 (24:A1604)
  • [5] V. Ganapathy Iyer, On effective sets of points in relation to integral functions, Trans. Amer. Math. Soc. 42 (1937), 358-365; correction, ibid. 43 (1938), 494. MR 1501926
  • [6] W. K. Hayman, Meromorphic functions, Oxford Math. Monographs, Clarendon Press, Oxford, 1964. MR 29 #1337. MR 0164038 (29:1337)
  • [7] A. J. MacIntyre, Wiman's method and the ``flat regions'' of integral functions, Quart. J. Math. Oxford Ser. (2) 9 (1938), 81-88.
  • [8] A. Pfluger, On analytic functions bounded at the lattice points, Proc. London Math. Soc. 42 (1936), 305-315.
  • [9] Keith Phillips, The maximal theorems of Hardy and Littlewood, Amer. Math. Monthly 74 (1967), 648-660. MR 35 #6788. MR 0215953 (35:6788)
  • [10] B. A. Taylor, Some locally convex spaces of entire functions, Proc. Sympos. Pure Math., vol. 11, Amer. Math. Soc., Providence, R. I., 1968, pp. 431-467.
  • [11] G. Valiron, Sur les variations du module des fonctions entières ou méromorphes, C. R. Acad. Sci. Paris 204 (1937), 33-35.
  • [12] J. M. Whittaker, On the ``flat'' regions of integral functions of finite order, Proc. Edinburgh Math. Soc. 2 (1930), 111-128.
  • [13] -, On the fluctuation of integral and meromorphic functions, Proc. London Math. Soc. 37 (1934), 383-401.
  • [14] -, Interpolatory function theory, Cambridge Tracts in Math., no. 33, Cambridge Univ. Press, New York, 1935.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 46.30, 30.00

Retrieve articles in all journals with MSC: 46.30, 30.00


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0290084-3
Keywords: Entire function, sufficient set, Fourier transform, subharmonic function
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society