The occurrence of almost periodic oscillations or waves is actually much more common than for periodic ones. It is our aim to present in this book, which we would like to serve as a graduate textbook on oscillations and waves, the basic facts related to the almost periodic case. We have taught this topic for several years and have received positive feedback from students.
In order to achieve our goal of presenting the basic facts related to almost periodic oscillations and waves, we will need some preliminaries, which will be treated in Chapter 2. These include the concept of metric space and some related topics. These introductory concepts will enable us to provide a solid mathematical foundation for the theory of several classes of almost periodic functions and their applications in physics and engineering.
A unified treatment of various classes of almost periodic functions requires some basic theory of metric spaces and related topics. The concept of metric spaces was introduced at the beginning of the twentieth century by the French mathematician Maurice Fréchet and has rapidly become one of the most useful tools in building modern analysis (including classical analysis and its development leading to functional analysis, both linear and nonlinear).
It is really a very simple task to define the concept of a metric space. It is likely that this simplicity in formulating the definition has contributed substantially to the diffusion of this concept in modern mathematics.
In Section 2.3, we defined the almost periodic functions in Bohr’s sense and stated some properties that follow more or less easily from their definition (see Remark 2.13 to Proposition 2.14 of Section 2.3). We shall briefly review those properties and provide some details in cases requiring clarification.
The Fourier analysis implies the study of such concepts as Fourier series (which constitutes a special case of the general concept of trigonometric series), Fourier transforms, Fourier integrals, and other related topics. It is well known that these concepts and related methods play a central role in investigating such phenomena as oscillations and waves. They are also relevant in systems theory and other branches of applied science. Fourier analysis also has been for almost two centuries one of the most powerful engines in advancing real analysis, approximation theory, and other fields.
The original concept of a Fourier series preoccupied not only Fourier and his contemporaries but many other mathematicians and scientists, the name of Euler being one of the first to be mentioned.
This concept of almost periodicity also represents an illustration of the more general situation when the Banach space X is substituted by a locally convex space. Namely, we shall deal with the dual of X, denoted by X*, and the local convex space Xw with the same elements as X but a different type of convergence than the norm convergence in X.
Let us briefly discuss these concepts before we formulate the definition of weak almost periodicity.
Of course, we want to take advantage of what we already know about linear systems and try to extend this knowledge to the quasilinear case (6.1).
We shall dwell particularly on the case A(t) = const., where the treatment can be carried out in a satisfactory manner. Proposition 5.13 offers a good example concerned with the linear case that can be used to obtain a similar result for associated quasilinear systems of the form (6.1). More precisely, the following result is obtained.
It is a well-known feature of the theory of partial differential equations that a unique solution is determined by prescribing additional conditions, usually initial conditions or boundary value conditions. Such conditions will appear in the subsequent discussion.
Constantin Corduneanu
Backmatter
Metadaten
Titel
Almost Periodic Oscillations and Waves
verfasst von
Constantin Corduneanu
Copyright-Jahr
2009
Verlag
Springer New York
Electronic ISBN
978-0-387-09819-7
Print ISBN
978-0-387-09818-0
DOI
https://doi.org/10.1007/978-0-387-09819-7
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