Almost Periodic and Almost Automorphic Functions in Abstract Spaces

It is known that a periodic function is a function that repeats its values at regular intervals. But the sum of two periodic functions is not always a periodic function. For example, $f(x)=\sin (x)+\sin (\sqrt {2}x)$ is not periodic.

In mathematics, an almost periodic function is, loosely speaking, a function whose value is approximately repeated when its argument is increased by properly selected constants (called almost periods) and the above mentioned example is an almost periodic function.

In 1925–1926, Harald Bohr published a series of foundational articles on almost periodic functions. Formally speaking, an almost periodic function is a continuous function $f:\mathbb{R}\to \mathbb{C}$ such that for every $\epsilon >0$, there are infinitely many $\tau =\tau (f,\epsilon )$ satisfying $\sup _{t\in \mathbb{R}}|f(t+\tau )-f(t)|<\epsilon$ such that the size of the gaps between successive $\tau$’ s is bounded. Each such $\tau (f,\epsilon )$ is said to be an $\epsilon$-period.

In 1961, Solomon Bochner 2 (see also Bochner 3 in 1964) introduced “a weakened concept of almost periodicity” (defined below) for which he said “we will designate this weakened concept as almost automorphy because we have been encountering it first, and on several occasions then, in the (differential geometric) study of automorphic functions on real and complex manifolds.”

In the following decades, the theory was extended to functions with values in a Banach space, with all the results collected together in the books: L. Amerio and G. Prouse, Almost Periodic Functions and Functional Equations, Van Nostrand Reinhold, New York, 1971 and C. Corduneau, Almost Periodic Functions, Chelsea Publishing Co., New York, 1989, and to functions with values in locally convex spaces by G. M. N’Guérékata in his 1980 PhD thesis 5.

The singular event which ignited a surge of interest in the theory of almost automorphic functions and applications to evolution equations was the 2001 publication of Gaston N’Guérékata’s Almost Automorphic and Almost Periodic Functions in Abstract Spaces (Kluwer Academic / Plenum, 2001).

This second edition of the book represents a remarkable effort in laying down the foundation of the theory of almost automorphic functions in Banach spaces and the theory of almost periodic functions in locally convex and nonlocally convex spaces and their applications to evolution equations.

The publication of the first edition of this book aroused widespread interest in the theory of almost automorphic functions and its applications to evolution equations. As a consequence, it was one of the most cited analysis books of the year 2001. In the years to follow, several generalizations of almost automorphic functions were introduced in the literature, including the study of almost automorphic sequences. In addition, the interplay between almost periodicity and almost automorphy has been explored for the first time in the context of operator theory, complex variable functions, and harmonic analysis methods. Readers will welcome the second edition of this book not only because it clarifies and improves on what was in the first edition, but because it presents new results, methods, and references. Also, this edition answers the fundamental question: “what is the number of almost automorphic functions that are not almost periodic in the sense of Bohr?” Open problems for almost periodic and almost automorphic functions with values in nonlocally convex spaces ($p$-Fréchet spaces) are also presented. For example, one major problem discussed is the lack of an established theory of integration of continuous functions with values in nonlocally convex spaces.

Recall here that a Banach space is a vector space $X$ over $\mathbb{C}$, endowed with a norm $\|\cdot \|$, such that with respect to the metric $d(x, y)=\|x-y\|$, $X$ is a complete metric space. That is every Cauchy sequence in $X$ is convergent. Also, an abstract evolution equation is a differential law of the development (evolution) in time of a system. More precisely, it is a partial differential equation with respect to the time variable $t\ge 0$ and the abstract spatial variable $x\in X$.

Chapter 1 of the second edition presents some basic concepts and results in Functional Analysis (without proofs). This chapter includes background material on topological vector spaces and operators which is used later in the text.

Chapter 2 gives a lovely introduction to the theory of Bochner–almost automorphic functions with values in a Banach space $X$. More precisely, it deals with continuous functions $f:\mathbb{R}\to X$ such that for every sequence of real numbers $(s'_n)$, there exists a subsequence $(s_n)$ such that

$$\begin{equation*} \lim _{m\to \infty }\lim _{n\to \infty }f(t+s_m-s_n)=f(t) \end{equation*}$$

for each $t\in \mathbb{R}$. This is equivalent to saying that a continuous function $f:\mathbb{R}\to X$ is said to be almost automorphic if for every sequence of real numbers $(s'_n)$, there exists a subsequence $(s_n)$ such that $g(t)\coloneq \lim _{n\to \infty }f(t+s_n)$ exists for each $t\in \mathbb{R}$ and

$$\begin{equation*} \lim _{n\to \infty }g(t-s_n)=f(t) \end{equation*}$$

for each $t\in \mathbb{R}$. The function $g$ above is measurable but not necessarily continuous. When the convergence above is uniform in $t\in \mathbb{R}$, one can show that the function is almost periodic in the sense of Bohr. The function $f(t)=\sin \left(\frac{1}{2+\cos t + \cos \sqrt {2} t}\right)$ is a typical example of an almost automorphic function which is not almost periodic.

Chapter 2 and the two that follow present basic properties of almost automorphic functions in a unified, homogeneous, and masterly way which lays the groundwork for applications to differential equations and dynamic time scales. These chapters should help researchers fill gaps in the theory and develop further extensions and applications in various fields.

Since it is hard to write down examples of almost automorphic functions, for a long time people wondered “how many” almost automorphic functions are not almost periodic. The answer was given by Z-M Zheng, H-S Ding and G.M. N’Guérékata in their remarkable 2013 paper which proves that the collection of all almost periodic functions is a set of first category in the space of all almost automorphic functions. This means that almost automorphic functions exist in plentitude. This result constitutes a major reason to study almost automorphic functions and consequently increases the interest in this book.

Another fundamental question is the following: if a function $f:\mathbb{R}\to X$ is almost automorphic, when is the integral $F(t)=\int _{0}^{t}f(\xi )d\xi$ also almost automorphic? In the case of almost periodicity, the answer was given by Kadets 4. It states that a necessary and sufficient condition is that $X$ does not contain any subspace isomorphic to $c_0$, the Banach space of all numerical sequences $(u_n)$ such that $\lim _{n\to \infty }u_n=0$, equipped with the supnorm. B. Basit 1 was able to extend Kadets’s theorem to the almost automorphic case. It’s too bad that the author does not present the proof of the important result of Basit in 1. He gives a nice proof of this result in the case of a general Banach space $X$. This result states that if a function $f:\mathbb{R}\to X$ is almost automorphic, then the integral $F(t)$ is almost automorphic if and only if the range of $f(t)$ is relatively compact in $X$.

Chapter 7 of the book deals with semigroups of bounded linear operators that behave like almost automorphic functions at infinity. Some of their topological and asymptotic properties based on the Nemytskii and Stepanov theory of dynamical systems are included. It is unfortunate that the author does not give a more in-depth presentation of the material in Chapter 7, specifically the study of asymptotically almost automorphic motions of dynamical systems and possible stability in the sense of Poisson motion. However, the exposition of this topic may lead to further research in this direction.

Chapters 8 and 9 deal with almost periodic functions with values in locally convex spaces and with values in non-locally convex spaces. For example, in their 2007 paper in the Global Journal of Pure and Applied Mathematics, the reviewer and G.M. N’Guérékata investigated the theory of Bohr–almost periodic functions in $p$-Fréchet spaces, $0<p<1$, which are nonlocally convex spaces. More precisely, for $0<p<1$, a vector space $X$ over $\mathbb{R}$ or $\mathbb{C}$ is called a $p$-Fréchet space, if it is endowed with a so-called $p$-norm $\|\cdot \|$ and $X$ is a complete metric space with respect to the metric $d(x, y)=\|x-y\|$. Recall that a $p$-norm satisfies the following conditions : $\|x+y\|\le \|x\|+\|y\|$, $\|x\|=0$ if and only if $x=0$, $\|\lambda x\|=|\lambda |^{p}\|x\|$.

The challenging question was to start with an appropriate definition. The authors were able to show that several results in the locally convex spaces setting hold in $p$-Fréchet spaces, $0<p<1$. However, due to the geometry of $p$-Fréchet spaces, $0<p<1$, especially because the Hahn–Banach theorem does not hold in such spaces, the fundamental theorem of calculus fails to be true. This implies the nonexistence of mean-values of such functions, a very important property of almost periodic functions with values in Banach spaces, which constitutes an obstacle in studying differential equations in nonlocally convex spaces. These open problems are presented in Chapter 9 of the book.

Chapters 10 and 11 discuss the applications of the previous results to differential equations in finite and infinite dimensional spaces.

An advantage of this second edition is that it corrects and gives a nice proof of the important result by D. Bugajewski and G.M. N’Guérékata stating that if $E$ is a Fréchet space, then the space of almost periodic functions $f: \mathbb{R}\to E$ is also a Fréchet space.

The Appendix is also a significant strength of the book. In particular, it presents a useful and simplified graph, showing the various relations between the classes of functions studied.

This book can serve as a reference for seminars and research on almost automorphy and almost periodicity in abstract spaces. It is accessible to anyone who is familiar with graduate-level functional analysis and operator theory and is easy to read. The exposition is clear and the proofs are given in detail. At the end of each chapter, there is a bibliography for the chapter content that an interested reader can use to further explore topics of interest. Also, the open problems presented in Chapter 9 on almost periodic and almost automorphic functions with values in nonlocally convex spaces ($p$-Fréchet spaces) are a unique asset of this second edition.