Multisummability in Carleman ultraholomorphic classes by means of nonzero proximate orders

Abstract

We introduce a general multisummability theory of formal power series in Carleman ultraholomorphic classes. The finitely many levels of summation are determined by pairwise comparable, nonequivalent weight sequences admitting nonzero proximate orders and whose growth indices are distinct. Thus, we extend the powerful multisummability theory for finitely many Gevrey levels, developed by J.-P. Ramis, J. Écalle and W. Balser, among others. We provide both the analytical and cohomological approaches, and obtain a reconstruction formula for the multisum of a multisummable series by means of iterated generalized Laplace-like operators.

Summability of formal solutions for some generalized moment partial differential equations

“…Proof. The first part is a direct consequence of Proposition 4.15, [9]. The second part is a direct consequence of the definition of the formal Borel transform (Definition 7) and the moment differentiation (Definition 9).…”

“…Condition (11) has already been used and justified in [9] (see Lemma 4.10, Remark 4.11 and Remark 4.12 in [9]), in order to obtain convolution kernels for multisummability. Definition 6.…”

On the Construction of Large Algebras Not Contained in the Image of the Borel Map

“…On the one hand, the study of the problem of algebrability with respect to this product might be a quite natural question. Moreover this product has become important very recently by the development of a convenient theory of multisummability of formal power series, see [13,Chapter 4] and [14]. Concerning these recent insights, in a private communication Prof. J. Sanz has told the authors the following explanations.…”

Multisummability of formal solutions for a family of generalized singularly perturbed moment differential equations

“…Recent results on generalized multisummability of formal power series concerning different (comparable and nonequivalent) levels associated with ultraholomorphic classes achieved in [6] are applied in the present study to achieve asymptotic properties of the solutions to a singularly perturbed moment differential equation. That concept of multisummability as long as previous results achieved by the authors in [13] have been the key points used to describe generalized multisummability of the formal solution of the main equation (1).…”