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Transactions of the American Mathematical Society

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Regularly varying functions and convolutions with real kernels


Author: G. S. Jordan
Journal: Trans. Amer. Math. Soc. 194 (1974), 177-194
MSC: Primary 40E05; Secondary 30A64
DOI: https://doi.org/10.1090/S0002-9947-1974-0342898-0
MathSciNet review: 0342898
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Abstract: Let $ \phi $ be a positive, measurable function and k a real-valued function on $ (0,\infty ),k \in {L^1}(dt/t)$. We give conditions on $ \phi $ and k sufficient to deduce the regular variation of $ \phi $ from the assumption that

$\displaystyle \alpha = \mathop {\lim }\limits_{x \to \infty } \frac{1}{{\phi (x... ...\frac{x}{2}} \right)} \;\frac{{dt}}{t}\;{\text{exits}}\;(\alpha \ne 0,\infty ).$

The general theorems extend in certain ways results of other authors and yield a new theorem on the relation between the radial growth and zero-distribution of those entire functions which are canonical products of nonintegral order with negative zeros.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1974-0342898-0
Keywords: Slowly varying, regularly varying, convolution, tauberian condition, integral equation, entire function
Article copyright: © Copyright 1974 American Mathematical Society

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