Log-convex sequences and nonzero proximate orders

TL;DR

This paper explores the relationship between log-convex sequences, nonzero proximate orders, and summability methods in ultraholomorphic classes, providing characterizations and construction techniques using regular variation theory.

Contribution

It offers new characterizations of nonzero proximate orders and demonstrates how to construct well-behaved strongly regular sequences from them.

Findings

Characterizations of nonzero proximate orders.

Construction methods for strongly regular sequences.

Connections between regular variation and summability properties.

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.