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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A limitation theorem for absolute summability


Author: Godfrey L. Isaacs
Journal: Proc. Amer. Math. Soc. 29 (1971), 47-54
MSC: Primary 40.40
DOI: https://doi.org/10.1090/S0002-9939-1971-0277955-3
MathSciNet review: 0277955
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Abstract: Let $ A(u)$ be of bounded variation over every finite interval of the nonnegative real axis, and let $ \smallint _0^w{e^{ - us}}dA(u)$ be summable $ \vert C,k\vert$ for a given integer $ k \geqq 0$ and a given s whose real part is negative. Then it is known that the function $ R(k,w) = (1/\Gamma (k + 1)) \cdot \smallint _w^\infty {(u - w)^k}dA(u)$ (which certainly exists in the $ \vert C,k\vert$ sense by a well-known summability-factor theorem) satisfies $ {e^{ - ws}}{w^{ - k}}R(k,w) = o(1)\vert C,0\vert(w \to \infty )$ . In this paper we extend the above result by showing that if the hypotheses are satisfied with k fractional, then $ {e^{ - ws}}{w^{ - k}}R(k + \delta ,w) = o(1)\vert C,0\vert$ for each $ \delta > 0$ and that this is best possible in the sense that $ \delta $ may not be replaced by 0.


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DOI: https://doi.org/10.1090/S0002-9939-1971-0277955-3
Keywords: Laplace-Stieltjes integral, Cesàro summability, summable $ \vert C,k\vert$
Article copyright: © Copyright 1971 American Mathematical Society