×

Espaces de Marcinkiewicz. Corrélations. Mesures, Systèmes dynamiques. Introduction de J. Bass. (Marcienkiewicz spaces. Correlations. Measures, dynamical systems). (French) Zbl 0617.46034

Paris etc.: Masson, VII, 249 p. FF 300.00 (1987).
Contents: J. Bass, Introduction; Kh. Vo Khac, Fonctions et distributions stationnaires. Application a l’étude des solutions stationnaires d’équations aux dérivées partielles; J. P. Bertrandias, Analyse harmonique dans l’espace de Marcinkiewicz; M. Mendès-France, Corrélations en théorie des nombres; Pham Phu Hien, Mesures asymptotiques; J. Couot, Systèmes dynamiques déterministes. Densités invariantes pseudo-aléatoires sur l’intervalle [0,1]; J. Dhombres, Moyennes.
The book is formed by a rather free connection of six chapters written by different authors and devoted to the theory of means with applications and to the investigation of properties of functions called chaotic, turbulent, stationary and pseudo-random. The motivation of these questions originate in certain physical processes and consequentional problems in the theory of partial differential equations. The first chapter introduces the (time) mean of a function f which maps \(R^ n\) into a Hilbert space H with the norm \(| \cdot |\) and the scalar product (.,.) by \(\lim_{j\to \infty}M_ j(f)\), where \(M_ j(f)=mes(j\Gamma)^{-1}\int_{j\Gamma}f(t)dt\), \(\Gamma\) is a (fixed) bounded convex neighbourhood of the origin in \(R^ n\) and mes stands for the Lebesgue measure. For \(p>0\) the Marcinkiewicz space \({\mathfrak M}^ p\) consists of all functions f such that \(\limsup_{j\to \infty}M_ j(| f|^ p)<\infty\). A function \(f\in {\mathfrak M}^ 2\) translations of which form continuous mappings is called stationary if for every \(a\in R^ n\) there exists the mean \(\gamma\) (a) of the scalar product of f(t) and of its translation \(f(t+a)\); \(\gamma\) is then called the correlation function of f. The first chapter serves in certain sense as a functional analytical basis for the book.
For the purpose of studying stationary functions the space \({\mathfrak M}^ 2\) is too large. Appropriate subspace is formed by functions called \({\mathfrak M}^ 2\)-regular, which do not possess big values on large sets, exactly (in one-dimensional case) for any subset \(E\subset R\) of density zero \(\lim_{T\to \infty}(2T)^{-1}\int | f(t)|^ 2dt\) with the integration over \(E\cap [-T,T]\). This and related subspaces are studied in the second chapter.
In the third chapter functions of discrete time are considered. This approach is justified by the fact that the basic pseudo-random functions are step functions whose means lead to the expressions of the form \(\lim_{N\to \infty}\frac{1}{N}\sum^{N}_{1}\). Constructions of pseudo-random functions utilize certain properties of sequences equidistributed modulo 1. Van der Corput’s theorem, which declares that a sequence \(\{u_ n\}\) is equidistributed modulo 1 if so is \(\{u_{n+h}- u_ n\}\) for every integer \(h\neq 0\), can be translated into the terms of pseudo-random functions and correlation functions. Here the theorem is extended in the sense that instead of all non-zero integers h a subset defined through the means and correlations can be considered. Characterizations and examples of such subsets are given.
The next chapter brings an extended definition of the asymptotic measure \(\mu\) of a function f of real variable with values in a topological space \(E: \mu\) is such that \(\int_{E}h(x)d\mu (x)=\lim_{T\to \infty}\frac{1}{2T}\int^{T}_{-T}h(f(t))dt\). Conditions on functions admitting asymptotic measures are given for \(E=R^ n\). The notion of asymptotic measure is extended for semigroups of transforms of E and the results are applied to the investigation of the asymptotic behaviour of solutions to evolution equations.
The author of the fifth chapter deals with dynamical systems resulting from transforms of the interval [0,1] given by piecewise continuous and monotone functions. For such systems asymptotical behaviour, stability, invariant measures and entropy are studied. One of important conclusions of this chapter is that there exist essential relations among the notion of the chaotic phenomenon, the structure of pseudo-random functions and the nature of solutions of dynamical systems.
The last chapter deals with general theory of means based on certain system of axioms. Starting from the usual types of means he comes to a general mean operator acting on function spaces. The results are applied to almost periodic functions and to the so called Reynolds operators.
The book is equipped by an expert preface of J. Bass, the author of monograph ”Fonctions de corrélation, fonctions pseudo-aléatoires et applications” (1984; Zbl 0557.76053), which can serve as an introduction to studying the present book. A detailed index makes the reading easier unlike numerous misprints. The book is not quite homogeneous although the authors took evidently some pains to unify the notations.
Reviewer: J.Rákosník

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
76Fxx Turbulence
76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics
37A99 Ergodic theory
35B40 Asymptotic behavior of solutions to PDEs
28D10 One-parameter continuous families of measure-preserving transformations

Citations:

Zbl 0557.76053