A limitation theorem for absolute summability

Isaacs, Godfrey L.

doi:10.1090/S0002-9939-1971-0277955-3

A limitation theorem for absolute summability
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by Godfrey L. Isaacs PDF

Proc. Amer. Math. Soc. 29 (1971), 47-54 Request permission

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 $|C,k|$ 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 $|C,k|$ sense by a well-known summability-factor theorem) satisfies ${e^{ - ws}}{w^{ - k}}R(k,w) = o(1)|C,0|(w \to \infty )$ . In this paper we extend the above result by showing that if the hypotheses are satisfied with k fractional, then ${e^{ - ws}}{w^{ - k}}R(k + \delta ,w) = o(1)|C,0|$ 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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