Strongly regular sequences and proximate orders

TL;DR

This paper investigates the relationship between strongly regular sequences, proximate orders, and summability kernels in ultraholomorphic classes, establishing conditions under which these concepts are equivalent and clarifying their growth properties.

Contribution

It provides a simple condition to associate a proximate order with a strongly regular sequence and shows the equivalence of key growth indices in ultraholomorphic classes.

Findings

Established a condition linking proximate orders to strongly regular sequences.

Proved the equality of the growth index and the order of quasianalyticity.

Characterized strongly regular sequences via regular variation.

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 (Summability in general Carleman ultraholomorphic classes, J. Math. Anal. Appl. 430 (2015), 1175--1206). We study several open questions related to the existence of kernels of summability constructed by means of analytic proximate orders. In particular, we give a simple condition that allows us to associate a proximate order with a strongly regular sequence. Under this assumption, and through the characterization of strongly regular sequences in terms of so-called regular variation, we show that the growth index $\gamma(\mathbb{M})$ defined by V.Thilliez (Division by flat ultradifferentiable functions and sectorial extensions, Results Math. 44 (2003), 169--188) and the order of quasianalyticity $\omega(\mathbb{M})$ introduced by the second author (Flat functions in Carleman ultraholomorphic classes via proximate orders, J. Math. Anal. Appl. 415 (2014), no. 2, 623--643) are the same.