About this ebook
The treatment, which is based on Bohr's lectures, starts with an introduction that leads to discussions of purely periodic functions and their Fourier series. The heart of the book, his exploration of the theory of almost periodic functions, is supplemented by two appendixes that cover generalizations of almost periodic functions and almost periodic functions of a complete variable.
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Almost Periodic Functions - Harald Bohr
INTRODUCTION
1. In planning the lectures which form the content of this book I had to make a choice which at first confronted me with a certain difficulty. Since the amount of time in which I had to give the lectures was rather limited, I had to decide in advance whether I wanted to give a more survey-like report on the whole theory or whether I should present only one part - but an essential one - of the theory, this, however, in fullest detail and with complete proofs. I decided for the latter alternative, holding the opinion that the lectures will be more permanently profitable for those attending if they are given the opportunity to assimilate the subject matter without haste - even if this means that the amount of material covered has to be somewhat reduced. The reduction of material was done mainly by dealing only with functions of a real variable, and besides only with continuous functions throughout.
2. In order to make up to some extent for the aforementioned reduction of material I have added two appendices to the lectures proper, as has already been said in the preface. The first of these deals with the generalization of the theory of almost periodic functions (along the lines of Lebesgue’s theory of the integral), while the second gives an account of the theory of almost periodic functions of a complex variable. This latter theory by the way, which has grown out of the theory of Dirichlet’s series, formed the author’s original starting point for the entire investigation.
For the reader who wishes to go deeper into the theory there is a short bibliography at the end, listing some of the pertinent works on the subject.
3. Before starting with the presentation of the theory proper we make some preliminary remarks concerning the problems taken up in the theory of almost periodic functions. Later on, of course, there will be given a detailed and precise formulation of these problems. Generally speaking we can say that the main problem of the theory consists in finding those functions f(x) of the real variable x which are defined for – ∞ < x < ∞ and which can be resolved into pure vibrations. This statement contains several words whose meanings must first of all be defined. What are pure vibrations,
and what is to be understood by the word resolved?
At this place I shall not go any deeper into the concept of resolution, which is most closely connected with the whole structure of the theory. However, I shall now explain precisely what is meant by a pure vibration.
4. As long as only real functions are considered, the term pure vibration
applies to such functions as cosx, sin x or more generally α cosx + β sinx; these functions are periodic of period 2π; if arbitrary periods are admitted, then pure vibration
will stand for a function of the form α cosλx + β sinλx. In the following it will be more convenient for formal reasons to consider not real, but complex functions of the real variable x. By a pure vibration is then meant any function of the form aeiλx = a(cosλx + i sinλx), where a denotes an arbitrary complex number and λ an arbitrary real number. If a is written in the form a = |a|eiv, the function takes on the form |a|eiveiλx. Here |a| gives the amplitude, v the phase and λ the frequency of the vibration. (The period of the vibration is (for λ ≠ 0) 2π/|λ|). Of these numbers, |a| and λ are the most important; frequently |a|² is considered instead of |a| itself.
5. The theory of almost periodic functions now deals with the following problem: Which functions f(x) can be resolved into pure vibrations on – ∞ < x < ∞ i.e., are re-presentable
by a trigonometric series of the form ? (We are here concerned only with series having at most denumerably many elements, or physically speaking, with functions whose spectrum
is a pure line spectrum; the theory of the Fourier integral , i.e., of the continuous spectra, is, therefore not, on principle, within the scope of our problem).
In the classical case where only harmonic vibrations are considered, i.e., vibrations of the form aeinαx, n = 0, ±1, ±2, … the corresponding question leads to the theory of the ordinary trigonometric series ; any periodic function of period p = 2π/|α| can, as is well known, be developed into such a series, the Fourier series of the function. The essentiel difference between this case and the above case where arbitrary vibrations aeiλx, – ∞ < λ < ∞ were considered, lies in the fact that in the latter case the set of the frequencies which occur is non-denumerable, whereas in the first case we are from the very start dealing only with denumerably many frequencies.
The lectures are divided into two parts. In the first-part we take up the classical case of harmonic vibrations aeinαx, choosing α = 1 for simplicity, so that we are dealing with the theory of ordinary Fourier series of period 2π This part is essentially of an introductory character and is intended mainly to make for an easier and deeper understanding of the following part; because of this, the exposition differs from the usual one in some respects.
In the second part there follows a treatment of the arbitrary vibrations aeiλx, which leads to the general theory of almost periodic functions and of their Fourier series,
6. Bohl’s theory of functions periodic in the more general sense
occupies an intermediate place between the theory of periodic and the theory of almost periodic functions. This theory corresponds to the case where only vibrations of the form
are considered, where α1, α2, … . αm are given linearly independent constants while n1, n2, …, nm run independently through all integers 0, ±1, ±2, … Every function periodic in the more general sense with the periods
p1 = 2π/|α1|, p2 = 2π/|α2|, …, pm = 2π/|αm| can be developed into a Fourier series of the form
The beautiful investigations by Bohl are closely related to the theory of almost periodic functions at many points; in a certain sense, however, these Bohl functions are nearer to the pure periodic functions than to the almost periodic ones, the reason being essentially that here, again, the basic system of frequencies is denumerable.
7. In the course of building up the theory, there presented itself, besides the Bohl functions just mentioned, another simple and important subclass, viz., the class of limit-periodic functions: These are the functions which can be developed into trigonometric aeries of the form where r assumes all rational values. For the sake of brevity, however, I could not in the present lectures enter into a discussion of either Bohl’s functions or the limit-periodic functions.
Also of necessity entirely omitted from the present publication are investigations such as those by Franklin and Bochner on almost periodic functions of several - even of denumerably many - variables; the investigations on the problem of distribution of values, a problem which has been attacked from different directions by Wintner and Jessen; also the investigations of differential equations with almost periodic coefficients, carried out mainly by Favard (who in this followed Bohl).
8. The theory of almost periodic functions was developed in its main features by the author in three rather long papers in the Acta Mathematica (Volumes 45, 46 and 47) under the common title Zur Theorie der Fast periodische Function en
; the first of these deals with the almost periodic functions of a real variable, while the third takes up the case of a complex variable.
In building the theory there was encountered a particular difficulty of intrinsic character; this is the decision on completeness
of the system of all functions eiλx. The original proof that this system is indeed a complete one (in a sense to be made precise later) was very complicated and involved many conclusions; its guiding idea, however, was a rather simple one; it was the idea, generally speaking, that the set of all purely periodic functions (with arbitrary periods) can be considered in a certain sense as being everywhere dense
within the general class of the almost periodic functions, so that from the very start it appeared quite possible to derive the validity of the completeness theorem by means of a limiting process from its well known validity for purely periodic functions (of a given period, i.e., for the system of harmonic vibrations einαx).
Wiener succeeded in finding a new proof of this fundamental theorem, much shorter than mine but, on the other hand, using results of the Lebesgue integral theory and the theory of Fourier integrals; whereas the original