Almost Periodic Functions (AMS Chelsea Publishing)
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From the Preface: 'The purpose of our book is to provide an overall view of all the basic features of almost periodic functions, in the various meanings this term has acquired in modern research as well as the many applications of such functions. This feature of the book should, of course, make it useful to those readers who are interested in the connections that exist between theory (pure!) of almost periodic functions and such areas of application as the theory of ordinary differential equations and partial differential equations. Several other areas of application are also indicated, together with an abundance of references'.From the Preface: 'The Bibliographical Notes at the end of almost every chapter have been added to in this second edition. Completely new is Chapter II.3, which is devoted to a relatively new class of almost periodic functions: random functions almost periodic in probability. It is certain that these functions will find many applications in the theory of stochastic functional equations'.
Introduction. Periodic functions, Fourier series Almost Periodic Functions: 1 Definitions and fundamental properties; 2 Two characteristic properties of almost periodic functions; 3 The Fourier series associated with an almost periodic function; 4 Convergence of Fourier series; 5 Summability of Fourier series; 6 Almost periodic numeric sequences and their connection with almost periodic functions; Bibliographical notes Almost Periodic Functions Depending on Parameters: 1 Almost periodic functions depending uniformly on parameters; 2 Almost periodic functions in the mean; 3 Random functions almost periodic in probability Analytic Almost Periodic Functions: 1 Preliminary theorems; 2 Elementary properties of analytic almost periodic functions; 3 The Dirichlet series associated with an analytic almost periodic function; 4 The behavior at infinity of analytic almost periodic functions in a half-plane; Bibliographical notes Almost Periodic Solutions of Ordinary Differential Equations: 1 The primitive of an almost periodic function; 2 Linear systems with constants coefficients; 3 Quasilinear systems; 4 An Amerio criterion for almost periodicity of bounded solutions; 5 The equation of non-linear oscillations; Bibliographical notes Almost Periodic Solutions of Partial Differential Equations: 1 Some properties of harmonic and polyharmonic functions; 2 Harmonic and polyharmonic almost periodic functions; 3 Almost periodic solutions of hyperbolic equations; 4 Almost periodic solutions of certain parabolic equations; Bibliographical notes Almost Periodic Functions with Values in Banach Spaces: 1 Definitions and general properties; 2 The Banach space of almost periodic functions; 3 The Fourier series attached to an almost periodic function with values in a Banach space. The approximation theorem; 4 Almost periodic functions in the Muckenhoupt sense and in the Stepanov sense; 5 Weakly almost periodic functions. The primitive of an almost periodic function; Bibliographical notes Almost Periodic Functions on Groups: 1 Elementary properties of almost periodic functions on groups. The existence of the mean; 2 Unitary representations of groups. The Fourier series associated with an almost periodic function; 3 The uniqueness and approximation theorems; 4 Almost periodic functions on topological groups; Bibliographical notes Appendix References Index.
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