Ilyashenko algebras based on transserial asymptotic expansions

We construct a Hardy field that contains Ilyashenko's class of germs at +infinity of almost regular functions found in [12] as well as all log-exp-analytic germs. This implies non-oscillatory behaviour of almost regular germs with respect to all log-exp-analytic germs. In addition, each germ in this Hardy field is uniquely characterized by an asymptotic expansion that is an LE-series as defined by van den Dries et al. [7]. As these series generally have support of order type larger than., the notion of asymptotic expansion itself needs to be generalized. (C) 2020 Elsevier Inc. All rights reserved.