Regularly varying functions and convolutions with real kernels
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- by G. S. Jordan
- Trans. Amer. Math. Soc. 194 (1974), 177-194
- DOI: https://doi.org/10.1090/S0002-9947-1974-0342898-0
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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 \[ \alpha = \lim \limits _{x \to \infty } \frac {1}{{\phi (x)}}\int _0^\infty {\phi (t)k\left ( {\frac {x}{2}} \right )} \;\frac {{dt}}{t}\;{\text {exits}}\;(\alpha \ne 0,\infty ).\] 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.References
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Bibliographic Information
- © Copyright 1974 American Mathematical Society
- 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