Strongly regular sequences and proximate orders

2015/10/20 by Jiménez-Garrido, Javier, Sanz, Javier · 1 citation
#26A12 #30D60 #40E05 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1510.05844

Abstract

Summability methods for ultraholomorphic classes in sectors, defined in terms of a strongly regular sequence \mathbbM=(M_p)_p∈ℕ0, 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(\mathbbM) defined by V.Thilliez (Division by flat ultradifferentiable functions and sectorial extensions, Results Math. 44 (2003), 169--188) and the order of quasianalyticity ω(\mathbbM) 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.

Cited by

• Dunkl derivative from moment differentiation

Related

• Summability in general Carleman ultraholomorphic classes
• Multisummability in Carleman ultraholomorphic classes by means of\n nonzero proximate orders
• Quasi-analyticity in Carleman ultraholomorphic classes
• Extension operators for some ultraholomorphic classes defined by sequences of rapid growth
• Surjectivity of the asymptotic Borel map in Carleman ultraholomorphic classes defined by sequences with shifted moments