R^(+)上的Loomis型定理及其应用  

Loomis type theorem on the half-line and its application

作　　者：简伟刚 丁惠生[1] Wei-Gang Jian;Hui-Sheng Ding

机构地区：[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

出　　处：《中国科学：数学》2023年第9期1241-1252,共12页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:11861037);江西省双千计划(批准号:jxsq2019201001)资助项目。

摘　　要：20世纪60年代Loomis(1960)给出了一个经典的结果:R上有界且一致连续函数谱集的可数性意味着其具有概周期性.然而,对于R^(+)上有界且一致连续的函数,即使其谱集是单点集,都不能保证其具有更弱的渐近概周期性.20世纪90年代末,Batty等(1998)给出了一个R^(+)上的Loomis型定理:R^(+)上完全遍历函数谱集的可数性意味着其具有渐近概周期性.对R^(+)上不具有完全遍历性的函数,是否有Loomis型结果?近二十多年,这方面一直没有本质性进展.本文通过建立渐近概周期函数的Kadets型定理,得到一个R^(+)上的Loomis型定理:R^(+)上有界且一致连续函数谱集的离散性意味着其具有遥远概周期性(比渐近概周期性略弱).本文的Loomis型定理完全去掉了Batty等结果中的遍历性假设,从某种意义上也是经典Loomis定理在R^(+)上的一个自然推广.并且,本文还将所得到的Loomis型定理应用到具有渐近概周期系数的Schrodinger方程,证明其仅存在更弱的遥远概周期解而没有渐近概周期解,从而说明对于某些偏微分方程,遥远概周期函数是其解的“自然函数类”这一有趣的现象.In the 1960s,Loomis(1960)published his famous theorem:a bounded and uniformly continuous function defined on R is almost periodic when its spectrum is at most countable.However,a bounded and uniformly continuous function defined on R^(+)is not necessarily asymptotically almost periodic even if the spectrum is a single point set.In the late 1990s,Batty et al.(1998)introduced a Loomis type theorem on R^(+):the countability of the spectrum of totally ergodic functions defined on R^(+)implies the asymptotically almost periodicity.Whereas,are there Loomis type results for functions which are not totally ergodic on R^(+)?It seems that there has been no essential progress in this regard in the past two decades.By establishing a Kadets type theorem for asymptotically almost periodic functions,we obtain our Loomis type theorem:the discreteness of the spectrum of bounded and uniformly continuous functions defined on R^(+)implies the remotely almost periodicity(weaker than the asymptotically almost periodicity).The Loomis type theorem in this paper removes the ergodicity assumption in the results of Batty et al.,and is a natural extension on R^(+)of the classical Loomis theorem.Moreover,we give an example of inhomogeneous Schr¨odinger equation with asymptotically almost periodic coefficients,whose solution is remotely almost periodic but not asymptotically almost periodic.This reveals an interesting phenomenon that the remotely almost periodic function is the“natural class”for solutions to some partial differential equations.

关 键 词：渐近概周期 遥远概周期 Loomis定理 CAUCHY问题 Carleman谱 

分 类 号：O175[理学—数学]