Closure theorems with applications to entire functions with gaps

Anderson, J. M.; Binmore, K. G.

doi:10.1090/S0002-9947-1971-0289141-6

Closure theorems with applications to entire functions with gaps
HTML articles powered by AMS MathViewer

by J. M. Anderson and K. G. Binmore

Trans. Amer. Math. Soc. 161 (1971), 381-400

DOI: https://doi.org/10.1090/S0002-9947-1971-0289141-6

PDF | Request permission

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.

References

J. M. Anderson and K. G. Binmore, Coefficient estimates for lacunary power series and Dirichlet series. I, II, Proc. London Math. Soc. (3) 18 (1968), 36–48; 49–68. MR 223576, DOI 10.1112/plms/s3-18.1.36
J. M. Anderson and K. G. Binmore, On entire functions with gap power series, Glasgow Math. J. 12 (1971), 89–97. MR 299790, DOI 10.1017/S0017089500001191
J. A. Clarkson and P. Erdös, Approximation by polynomials, Duke Math. J. 10 (1943), 5–11. MR 7813, DOI 10.1215/S0012-7094-43-01002-6
Albert Edrei, Gap and density theorems for entire functions, Scripta Math. 23 (1957), 117–141. MR 98829
W. H. J. Fuchs, On the closure of $\{e^{-t}t^{a_\nu }\}$, Proc. Cambridge Philos. Soc. 42 (1946), 91–105. MR 14501
Dieter Gaier, On the coefficients and the growth of gap power series, SIAM J. Numer. Anal. 3 (1966), 248–265. MR 204653, DOI 10.1137/0703019
Jean-Pierre Kahane, Travaux de Beurling et Malliavin, Séminaire Bourbaki, Vol. 7, Soc. Math. France, Paris, 1995, pp. Exp. No. 225, 27–39 (French). MR 1611202
J. Korevaar, A characterization of the sub-manifold of $C[a,b]$ spanned by the sequence $\{x_k{}^n\}$, Nederl. Akad. Wetensch., Proc. 50 (1947), 750–758 = Indagationes Math. 9, 360–368 (1947). MR 23368
T. Kővari, On the growth of entire functions of finite order with density conditions, Quart. J. Math. Oxford Ser. (2) 17 (1966), 22–30. MR 193235, DOI 10.1093/qmath/17.1.22
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 0146558
Paul Malliavin, Sur quelques procédés d’extrapolation, Acta Math. 93 (1955), 179–255 (French). MR 75297, DOI 10.1007/BF02392523
S. Mandelbrojt, Séries adhérentes, régularisation des suites, applications, Gauthier-Villars, Paris, 1952 (French). MR 0051893