Tauberian Theorems For Some Almost Periodic Type Functions And Their Applications

In 1960,Loomis in Ann.of Math.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 necessary asymptotically almost periodic even if the spectrum is a single point set.In addition,the similar situation is encountered for asymptotically almost periodic functions defined on R.Therefore,there are many substantial difficulties when one considers Loomis type theorem on the half-line or for the asymptotically almost periodic functions.Until the 1990s,Batty et al.proved that the countability of spectrum of totally ergodic functions defined on R+ implies asymptotically almost periodicity.Whereas,are there Loomis type results for functions without ergodicity?In the past two decades,there are no essential progress in this regard.The central content of this paper revolves around the above questions.We firstly prove the bounded primitive function of an asymptotically almost periodic function is remotely almost periodic(weaker than asymptotically almost periodic)if and only if there is not any closed linear subspace of X which is isomorphic to c0.On the basis of this theorem and its related results,we obtain many Tauberian results for a series of almost periodic type functions,including a Loomis type theorem on half-line:the discreteness of the spectrum of bounded and uniformly continuous functions defined on R+ implies remotely almost periodicity.Finally,we apply these theorems to some differential(integral)equations.This paper is divided into five chapters as follows.In Chapter 1,we introduce the relevant historical background and development.In Chapter 2,in addition to reviewing some basic concepts and terminology,we also introduce several new concepts and prove relevant properties.In Chapter 3,we establish a series of Kadets type theorems for kinds of almost periodic type functions by the classification of functions.The central content of this paper is Chapter 4.We introduce Tauberian theorems for kinds of functions by dividing on R and on R+.In Chapter 5,we study some equations(cf.Cauchy problems)by using our Tauberian theorems.Particularly,in§5.1,we give an example of inhomogeneous Schrodinger equation with asymptotically almost periodic coefficient,whose solution is remotely almost periodic but not asymptotically almost periodic.This not only reveals an interesting phenomenon that remotely almost periodic functions is the "natural class" for solutions to some partial differential equations,but also states our Loomis type theorem on R+ for asymptotically almost functions is well complement and perfection of the classical theory.