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Transactions of the American Mathematical Society

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A density theorem with an application to gap power series


Author: K. G. Binmore
Journal: Trans. Amer. Math. Soc. 148 (1970), 367-384
MSC: Primary 30.20
MathSciNet review: 0255776
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Abstract: Let N be a set of positive integers and let

$\displaystyle F(z) = \sum {{A_n}{z^n}} $

be an entire function for which $ {A_n} = 0(n \notin N)$. It is reasonable to expect that, if D denotes the density of the set N in some sense, then $ F(z)$ will behave somewhat similarly in every angle of opening greater than $ 2\pi D$. For functions of finite order, the appropriate density seems to be the Pólya maximum density $ \mathcal{P}$. In this paper we introduce a new density $ \mathcal{D}$ which is perhaps the appropriate density for the consideration of functions of unrestricted growth. It is shown that, if $ \vert I\vert > 2\pi \mathcal{D}$, then

$\displaystyle \log M(r) \sim \log M(r,I)$

outside a small exceptional set. Here $ M(r)$ denotes the maximum modulus of $ F(z)$ on the circle $ \vert z\vert = r$ and $ M(r,I)$ that of $ F(r{e^{i\theta }})$ for values of $ \theta $ in the closed interval I. The method used is closely connected with the question of approximating to functions on an interval by means of linear combinations of the exponentials $ {e^{ixn}}(n \in N)$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1970-0255776-9
Keywords: Gap power series, entire functions, closure, completeness and freedom, $ {\mathcal{L}^p}$ spaces, Beurling-Malliavin density, Pólya maximum density, Fourier transforms
Article copyright: © Copyright 1970 American Mathematical Society