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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

A note on Roe's characterization of the sine function


Author: Ralph Howard
Journal: Proc. Amer. Math. Soc. 105 (1989), 658-663
MSC: Primary 33A10; Secondary 42A38
MathSciNet review: 942633
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Abstract: Let $ {f^{(n)}}n = 0, \pm 1, \pm 2, \ldots $ be a sequence of complex valued functions on the real line with $ (d/dx){f^{(n)}} = {f^{(n + 1)}}$ and satisfying inequalities $ \vert{f^{(n)}}(x)\vert \leq {M_n}{(1 + \vert x\vert)^k}$ where as $ n \to \infty $ the growth conditions $ \underline {\lim } {M_n}{(1 + \varepsilon )^{ - n}} = 0$ and $ \underline \lim {M_{ - n}}{(1 + \varepsilon )^{ - n}} = 0$ hold for all $ \varepsilon > 0$. Then $ {f^{(0)}}(x) = p(x){e^{ix}} + q(x){e^{ - ix}}$ where $ p$ and $ q$ are polynomials of degree at most $ k$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1989-0942633-5
PII: S 0002-9939(1989)0942633-5
Article copyright: © Copyright 1989 American Mathematical Society