 Author

Jacob Korevaar
 Year
 2000
 Title
 Growth of power series with square root gaps
 Publisher
 s.n.
 Document type
 Working paper
 Faculty
 Faculty of Science (FNWI)
 Institute
 Kortewegde Vries Institute for Mathematics (KdVI)
 Abstract

For entire functions $f$ whose power series have Hadamard gaps with ratio $\alpha$ $(>1)$, Gaier has shown that the condition $f(x)\le e^x$ for $x\ge 0$ implies $f(z)\le C_\alpha e^{z}$ $(*)$ for all $z$. Here the result is extended to the case of square root gaps, that is, $f(z)=\sum b_{p_k}z^{p_k}$ with $p_{k+1}p_k\ge \alpha\sqrt{p_k}$ where $\alpha>0$. Smaller gaps cannot work. In connection with his proof of the general high indices theorem for Borel summability, Gaier had shown that square root gaps imply $b_n={\cal O}(e^{c\sqrt n}/n!)$. Having such an estimate, one can adaptPitt's Tauberian method for the restricted Borel high indices theorem to show that, in fact,$b_n\le c_\alpha\sqrt n/n!$ which implies $(*)$. The author also states an equivalent distance formula involving monomials $x^{p_k}e^{x}$ in $L^\infty(0,\infty)$.
 Language
 Undefined/Unknown
 Permalink
 http://hdl.handle.net/11245/1.418861
 Downloads
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