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Moment Summation Methods and Non-Homogeneous Carleman Classes

Aver Kiro

TL;DR

This work develops a comprehensive framework to study how spaces of smooth functions transform under the generalized Laplace transform $\mathcal{L}_{\gamma}$ by introducing non-homogeneous Carleman classes. It establishes precise local and global (analytic and non-analytic) transfer principles between Borel-type and Carleman-type classes, providing necessary and sufficient conditions for moment-summation applicability and extending Écalle’s quasianalytic continuation. The results unify and extend classical Nevanlinna-Beurling descriptions, deliver strong remainder and derivative estimates, and connect to multi-summability and resurgent theories. The framework yields practical tools for solving Euler-type ODEs and for analyzing multi-scale asymptotics in analytic and quasianalytic settings with concrete domain descriptions such as $\Omega_{\eta}(\gamma)$. Overall, the paper advances the theory of summation, extension, and analytic continuation in highly general Carleman-analytic contexts with broad applications to differential equations and resurgent analysis.

Abstract

We extend the classical theorems of F. Nevanlinna and Beurling by characterizing the image of various spaces of smooth functions under the generalized Laplace transform. To achieve this, we introduce and analyze novel non-homogeneous Carleman classes, which generalize the traditional homogeneous definitions. This characterization allows us to derive necessary and sufficient conditions for the applicability of moment summation methods within a given class of functions. Furthermore, we establish an extension of Écalle's concept of quasianalytic continuation and apply these results to the theory of multi-summability and Euler-type differential equations.

Moment Summation Methods and Non-Homogeneous Carleman Classes

TL;DR

This work develops a comprehensive framework to study how spaces of smooth functions transform under the generalized Laplace transform by introducing non-homogeneous Carleman classes. It establishes precise local and global (analytic and non-analytic) transfer principles between Borel-type and Carleman-type classes, providing necessary and sufficient conditions for moment-summation applicability and extending Écalle’s quasianalytic continuation. The results unify and extend classical Nevanlinna-Beurling descriptions, deliver strong remainder and derivative estimates, and connect to multi-summability and resurgent theories. The framework yields practical tools for solving Euler-type ODEs and for analyzing multi-scale asymptotics in analytic and quasianalytic settings with concrete domain descriptions such as . Overall, the paper advances the theory of summation, extension, and analytic continuation in highly general Carleman-analytic contexts with broad applications to differential equations and resurgent analysis.

Abstract

We extend the classical theorems of F. Nevanlinna and Beurling by characterizing the image of various spaces of smooth functions under the generalized Laplace transform. To achieve this, we introduce and analyze novel non-homogeneous Carleman classes, which generalize the traditional homogeneous definitions. This characterization allows us to derive necessary and sufficient conditions for the applicability of moment summation methods within a given class of functions. Furthermore, we establish an extension of Écalle's concept of quasianalytic continuation and apply these results to the theory of multi-summability and Euler-type differential equations.
Paper Structure (44 sections, 30 theorems, 315 equations, 6 figures, 1 table)

This paper contains 44 sections, 30 theorems, 315 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Smoothness of the generalized Laplace transform
  3. Main results
  4. A local result
  5. A global result in the non-analytic case
  6. A global result in the analytic case
  7. F. Nevanlinna and Beurling theorems
  8. Structure of this work
  9. Basic notions
  10. Notation
  11. Admissible weights and the sequence $\widehat{\gamma}$
  12. The functions $K$ and $E$
  13. The saddle point equation
  14. Asymptotic behavior of the functions $K$ and $E$
  15. Carleman type classes
  16. ...and 29 more sections

Key Result

Theorem 1.4

Let $\gamma$ be an admissible weight and let $M$ be a regular sequence. For any $\eta>1,$ we have Moreover, if the class $A(M,\gamma;0_{+})$ is quasianalytic (i.e., if $\sum M_{n}/M_{n+1}=\infty)$, then and the $(\gamma)$--summation is applicable in the class $B_{1/\eta}(M\gamma,\widehat{\gamma};0_{+})$.

Figures (6)

  • Figure 1.1: $\text{$\Omega_{1}\left(\gamma\right)$ for $\alpha=\frac{1}{2},\,1,\,2$}$
  • Figure 1.2: The disk $\Omega_{\eta}$
  • Figure 4.1: $\Gamma=\Gamma_{1}+\Gamma_{2}+\Gamma_{3}$
  • Figure 5.1: The set $V_{k,\eta,Q}$ for $J=[0,1]$
  • Figure 5.2: The set $U_{k,\eta,Q}$ for $J=[0,1]$
  • ...and 1 more figures

Theorems & Definitions (84)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Definition 1.7
  • Example 1.8
  • Lemma 1.9
  • Theorem 1.10
  • ...and 74 more