Tauberian theorems for k–summability with respect to an analytic germ

Carrillo, Sergio A.; Mozo-Fernández, Jorge; Schäfke, Reinhard

doi:10.1016/j.jmaa.2020.124174

Cited by 6 publications

(11 citation statements)

References 10 publications

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“…Integral transformations for the corresponding summability methods are still not available and it is an interesting problem to determine whether it is possible to extend our results to that setting. It is worth mentioning that after our current results, we have recently extended these Tauberian theorems for k-summability with respect to analytic germs, see [8].…”

Section: Introductionmentioning

confidence: 62%

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Summability in a monomial for some classes of singularly perturbed partial differential equation

Carrillo

2018

Preprint

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The aim of this paper is to continue the study of asymptotic expansions and summability in a monomial in any number of variables, as introduced in [4,15]. In particular, we characterize these expansions in terms of bounded derivatives and we develop Tauberian theorems for the summability processes involved. Furthermore, we develop and apply the Borel-Laplace analysis in this framework to prove the monomial summability of solutions of a specific class of singularly perturbed PDEs.

show abstract

Section: Introductionmentioning

confidence: 62%

“…This provides more examples of series that cannot be summed with the methods we have studied along this work. In fact, this has been done recently in [8] in the more general setting of k-summability in analytic germs and we refer the reader to this work for a complete proof of these facts.…”

Section: Given F ∈ O ′mentioning

confidence: 81%

Summability in a monomial for some classes of singularly perturbed partial differential equation

Carrillo

2018

Preprint

Self Cite

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show abstract

Section: Introductionmentioning

confidence: 64%

“…This provides more examples of series that cannot be summed with the methods we have studied along this work. In fact, this has been done recently in [7] in the more general setting of k-summability in analytic germs and we refer the reader to this work for a complete proof of these facts.…”

Section: Proposition 52 Iff ∈ E{x} αmentioning

confidence: 81%

Summability in a monomial for some classes of singularly perturbed partial differential equations

Carrillo¹

2021

Publicacions Matemàtiques

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The aim of this paper is to continue the study of asymptotic expansions and summability in a monomial in any number of variables, as introduced in [3, 15]. In particular, we characterize these expansions in terms of bounded derivatives and we develop Tauberian theorems for the summability processes involved. Furthermore, we develop and apply the Borel-Laplace analysis in this framework to prove the monomial summability of solutions of a specific class of singularly perturbed PDEs.

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“…But inequality (8) implies that a j n−k ≤ a j 0 . Therefore, if a = a 1 0 +• • •+ a d 0 , by hypothesis a > 0, and L(y k−1 ) n ≤ eak y k−1 k−1 .…”

Section: Let Us Check First Thatmentioning

confidence: 99%

Formal $P$-Gevrey series solutions of first order holomorphic PDEs

Torres¹,

Hurtado²

2020

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The goal of this work is to exhibit a Gevrey type, in an analytic function P , of formal power series solutions of some families of first order holomorphic PDEs. The approach is based on the classical majorant series technique by applying Nagumo norms joint with a division algorithm. Our main result recovers systematically many situations studied in the literature on the Gevrey type of formal solutions of these equations. We also provide a relation between Gevrey series in P and Gevrey series in several variables.

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Tauberian theorems for k–summability with respect to an analytic germ

Cited by 6 publications

References 10 publications

Summability in a monomial for some classes of singularly perturbed partial differential equation

Summability in a monomial for some classes of singularly perturbed partial differential equation

Summability in a monomial for some classes of singularly perturbed partial differential equations

Formal $P$-Gevrey series solutions of first order holomorphic PDEs

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