渐近周期函数的 Tauberian 定理及其在抽象 Cauchy 问题中的应用

数学物理学报 ›› 2023, Vol. 43 ›› Issue (6): 1699-1709.

渐近周期函数的 Tauberian 定理及其在抽象 Cauchy 问题中的应用

简伟刚^1,²(),龙薇^1,^*()

¹江西师范大学数学与统计学院南昌 330022
²豫章师范学院数学与计算机学院南昌 330103

收稿日期:2023-01-16 修回日期:2023-04-10 出版日期:2023-12-26 发布日期:2023-11-16
通讯作者: *龙薇，E-mail: lwhope@jxnu.edu.cn
作者简介:简伟刚，E-mail: yuzhu@jxnu.edu.cn
基金资助:
国家自然科学基金(11861037);江西省双千计划(jxsq2019201001);江西省自然科学基金重点项目(20212ACB201003)

Tauberian Theorem for Asymptotically Periodic Functions and Its Application to Abstract Cauchy Problems

Jian Weigang^1,²(),Long Wei^1,^*()

¹School of Mathematics and Statistics, Jiangxi Normal University, Nanchang 330022
²School of Mathematics and Computer, Yuzhang Normal University, Nanchang 330103

Received:2023-01-16 Revised:2023-04-10 Online:2023-12-26 Published:2023-11-16
Supported by:
NSFC(11861037);Two Thousand Talents Program of Jiangxi Province(jxsq2019201001);Jiangxi Provincial Natural Science Foundation(20212ACB201003)

摘要/Abstract

摘要：

周期函数的有界原函数是周期函数, 而渐近周期函数的有界原函数未必是渐近周期函数. 该文引入了缓慢周期函数的概念, 并证明了渐近周期函数的有界原函数是缓慢周期函数. 有趣的是, 缓慢周期函数恰好是一类特殊的 $\S$-渐近周期函数, 而 $\S$-渐近周期函数早在 15 年前就被引入且近年来被广泛研究. 在此基础上, 建立了渐近周期函数的 Tauberian 定理及两个相关 Tauberian 定理. 此外, 将所得 Tauberian 定理应用到非齐次抽象 Cauchy 问题, 得到了 Cauchy 问题的解具有 $\S$-渐近周期性的谱集判定定理. 该文建立的渐近周期函数的 Tauberian 定理和抽象 Cauchy 问题的谱集判定定理的结论虽然比渐近周期性略弱, 但彻底去掉了文献 [23] 中的遍历性假设. 最后, 构造了一个具体的 Cauchy 问题作为例子. 值得一提地是, 该 Cauchy 问题的非齐次项是渐近周期函数, 但它的解却不是渐近周期的而是 $\S$- 渐近周期的. 这说明了 $\S$-渐近周期函数是一些微分方程解的"自然"函数类.

关键词: 渐近周期, 缓慢周期, $\S$-渐近周期, 抽象 Cauchy 问题, Tauberian 定理, Beurling 谱

Abstract:

The bounded primitive of a periodic function is periodic, and the bounded primitive of an asymptotically periodic function is not necessarily asymptotically periodic. In this paper, we introduce the concept of slowly periodic functions and prove that the bounded primitive function of an asymptotically periodic function is slowly periodic. Interestingly, slowly periodic functions are just a special class of $\S$-asymptotically periodic functions, which were introduced 15 years ago and extensively studied in recent years. On this basis, a Tauberian theorem for asymptotically periodic functions and two related Tauberian theorems are established. Moreover, we apply our Tauberian theorems to the nonhomogeneous abstract Cauchy problem, and obtain the spectral condition under which the solution of Cauchy problem is $\S$-asymptotically periodic. In our Tauberian theorem for asymptotically periodic functions and its application to abstract Cauchy problem, we completely remove the ergodic assumption in [23] although the conclusions are slightly weaker than asymptotical periodicity. Finally, we construct a concrete Cauchy problem as an example. It is worth mentioning that the inhomogeneous term of this Cauchy problem is asymptotically periodic and its solution is $\S$-asymptotically periodic rather than asymptotically periodic. This demonstrates that $\S$-asymptotically periodic functions are the "natural class'' for solutions to some differential equations.

Key words: Asymptotically periodic, Slowing periodic, $\S$-asymptotically periodic, Abstract Cauchy problem, Tauberian theorem, Beurling spectrum

中图分类号:

O177.7

简伟刚, 龙薇. 渐近周期函数的 Tauberian 定理及其在抽象 Cauchy 问题中的应用[J]. 数学物理学报, 2023, 43(6): 1699-1709.

Jian Weigang, Long Wei. Tauberian Theorem for Asymptotically Periodic Functions and Its Application to Abstract Cauchy Problems[J]. Acta mathematica scientia,Series A, 2023, 43(6): 1699-1709.

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