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Generalization of characterizations of the trigonometric functions

  • JONG-HO KIM (a1), YUN-SUNG CHUNG (a1) and SOON-YEONG CHUNG (a2)
Abstract

Suppose that $P(D)$ is a linear differential operator with complex coefficients and $\{f_k\}_{k \in \mathbb{Z}}$ is a two-sided sequence of complex-valued functions on $\mathbb{R}$ such that $f_{k+1} \,{=}\, P(D)f_k$ with the following growth condition: there exist $a\,{\in}\,[0,1)$ and $N\,{\geq}\,0$ such that $|f_k(x)|\,{\leq}\,M_{|k|}\exp (N|x|^a)$, where the sequence $\{M_k\}_{k=0}^{\infty}$ satisfies that for any $\varepsilon\,{>}\,0$, the sequence $\{{M_k}/{(1+\varepsilon)^{k}}\}_{k=0}^{\infty}$ has a bounded subsequence. Then $f_0$ is an entire function of an exponential growth. This result is a generalization of those of Roe, Burkill and Howard.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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