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Transactions of the American Mathematical Society

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Closure theorems with applications to entire functions with gaps


Authors: J. M. Anderson and K. G. Binmore
Journal: Trans. Amer. Math. Soc. 161 (1971), 381-400
MSC: Primary 90.57
DOI: https://doi.org/10.1090/S0002-9947-1971-0289141-6
MathSciNet review: 0289141
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Abstract: In this paper we consider questions of completeness for spaces of continuous functions on a half line which satisfy appropriate growth conditions. The results obtained have consequences in the theory of entire functions with gap power series. In particular we show that, under an appropriate gap hypothesis, the rate of growth of an entire function in the whole plane is determined by its rate of growth along any given ray.


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  • [1] J. M. Anderson and K. G. Binmore, Coefficient estimates for lacunary power series and Dirichlet. I, Proc. London Math. Soc. (3) 18 (1968), 36-48. MR 36 #6624. MR 0223576 (36:6624)
  • [2] -, On entire functions with gap power series, Glasgow Math. J. (to appear). MR 0299790 (45:8838)
  • [3] J. A. Clarkson and P. Erdös, Approximation by polynomials, Duke Math. J. 10 (1943), 5-11. MR 4, 196. MR 0007813 (4:196e)
  • [4] A. Edrei, Gap and density theorems for entire functions, Scripta Math. 23 (1958), 117-141. MR 20 #5281. MR 0098829 (20:5281)
  • [5] W. H. J. Fuchs, On the closure of $ \{ {e^{ - t}}{t^{{a_\nu }}}\} $, Proc. Cambridge Philos. Soc. 42 (1946), 91-105. MR 7, 294. MR 0014501 (7:294b)
  • [6] D. Gaier, On the coefficients and the growth of gap power series, SIAM J. Numer. Anal. 3 (1966), 248-265. MR 34 #4492. MR 0204653 (34:4492)
  • [7] J.-P. Kahane, Travaux de Beurling et Malliavin, Séminaire Bourbaki 1961/62, Exposé 225 Secrétariat mathématique, Paris, 1962. MR 26 #3561; errata, MR 30 p. 1203. MR 1611202
  • [8] J. Korevaar, A characterisation of the sub-manifold of $ C[a,b]$ spanned by the sequence $ \{ {x^{{n_k}}}\} $, Nederl. Akad. Wetensch. Proc. Ser. A 50 (1947), 750-758 = Indag. Math. 9 (1947), 360-368. MR 9, 346. MR 0023368 (9:346b)
  • [9] T. Kövari, On the growth of entire functions of finite order with density conditions, Quart. J. Math. Oxford Ser. (2) 17 (1966), 22-30. MR 33 #1456. MR 0193235 (33:1456)
  • [10] A. F. Leont'ev, On the completeness of a system of powers on a semi-axis, Izv. Akad. Nauk SSSR Ser. Mat. 26 (1962), 781-792. (Russian) MR 26 #4080. MR 0146558 (26:4080)
  • [11] P. Malliavin, Sur quelques procédés d'extrapolation, Acta Math. 93 (1955), 179-255. MR 17, 724. MR 0075297 (17:724c)
  • [12] S. Mandelbrojt, Séries adhérentes, régularisation des suites, applications, Gauthier-Villars, Paris, 1952. MR 14, 542. MR 0051893 (14:542f)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1971-0289141-6
Keywords: Banach space, entire function, closure, approximation, power series
Article copyright: © Copyright 1971 American Mathematical Society

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