Javier Jiménez-Garrido — Page 2

arXiv2017

A Phragmén-Lindelöf theorem via proximate orders, and the propagation of asymptotics

We prove that, for asymptotically bounded holomorphic functions in a sector in $\mathbb{C}$, an asymptotic expansion in a single direction towards the vertex with constraints in terms of a logarithmically convex sequence admitting a nonzero proximate order entails asymptotic expansion in the whole sector with control in terms of the same sequence. This generalizes a result by A. Fruchard and C. Zhang for Gevrey asymptotic expansions, and the proof strongly rests on a suitably refined version of the classical Phragmén-Lindelöf theorem, here obtained for functions whose growth in a sector is specified by a nonzero proximate order in the sense of E. Lindelöf and G. Valiron.

math.CV↗

arXiv2015

Strongly regular sequences and proximate orders

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 $γ(\mathbb{M})$ defined by V.Thilliez (Division by flat ultradifferentiable functions and sectorial extensions, Results Math. 44 (2003), 169--188) and the order of quasianalyticity $ω(\mathbb{M})$ 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.

math.CA↗