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arXiv · 1607.08027

Log-convex sequences and nonzero proximate orders

Javier Jiménez-Garrido·Javier Sanz·Gerhard Schindl
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Abstract

Summability methods for ultraholomorphic classes in sectors, defined in terms of a strongly regular sequence $\mathbb{M}=(M_p)_{p\in\mathbb{N}_0}$, have been put forward by A. Lastra, S. Malek and the second author [1], and their validity depends on the possibility of associating to $\mathbb{M}$ a nonzero proximate order. We provide several characterizations of this and other related properties, in which the concept of regular variation for functions and sequences plays a prominent role. In particular, we show how to construct well-behaved strongly regular sequences from nonzero proximate orders. [1] A. Lastra, S. Malek and J. Sanz, Summability in general Carleman ultraholomorphic classes, J. Math. Anal. Appl. 430 (2015), 1175--1206.

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Javier Jiménez-Garrido, Javier Sanz, Gerhard Schindl. 2016-07-27. Log-convex sequences and nonzero proximate orders. https://doi.org/10.1016/j.jmaa.2016.11.069

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