A density theorem with an application to gap power series

Binmore, K. G.

doi:10.1090/S0002-9947-1970-0255776-9

A density theorem with an application to gap power series
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by K. G. Binmore PDF

Trans. Amer. Math. Soc. 148 (1970), 367-384 Request permission

Abstract:

Let N be a set of positive integers and let \[ F(z) = \sum {{A_n}{z^n}} \] 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 $|I| > 2\pi \mathcal {D}$, then \[ \log M(r) \sim \log M(r,I)\] outside a small exceptional set. Here $M(r)$ denotes the maximum modulus of $F(z)$ on the circle $|z| = r$ 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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