Kahane J.-P. Some Random Series of Functions

Cambridge University Press, 1994, -315 p.The origin of this book, whose first edition was published in 1968, is a series of notes by Paley and Zygmund that appeared in the 1930s, entitled 'On some series of functions' [167]. Paley and Zygmund, with the collaboration of Wiener in a subsequent paper, studied Fourier or Taylor series whose coefficients are independent random variables.
Applying random methods to harmonic analysis is an old idea, and it was the first aim of my book in 1968. In many circumstances it is hard or even impossible to find a mathematical object with some prescribed properties, and pretty easy to exhibit a random object which enjoys these properties almost surely. The classical example of this situation is the following theorem: no better condition than Riesz Fischer's, bearing on the absolute values of the coefficients of a trigonometric series, can ensure that it is a Fourier Lebesgue series.
A most beautiful example is the Pisier algebra, which solves completely an old problem of Y. Katznelson of finding a homogeneous Banach algebra of continuous functions on the circle, with the condition that not all continuous functions operate in this algebra, and not only analytic functions. There have been some difficult constructions due to M. Zafran. Pisier's solution is the algebra of continuous functions such that, changing signs randomly in their Fourier series, the resulting series represents almost surely a continuous function.
From the random point of view it often happens that the strange becomes natural (like a nowhere differentiable function, an almost every- where divergent series, a set of multiplicity which is independent over the rationals, and so on). That proves especially true in the kind of geometrical figures that B. Mandelbrot called fractals. Sets uncovered by random intervals, images of a given set by a random process, level sets of a random function or random field, were already studied in my book in 1968, with the help of some elementary Fourier analysis. I am doing a little more in the present edition, and I devote one chapter to an introduction to the geometrical part of the book. I have replaced the chapter on covering the circle by random arcs by an exposition of the work ofL. Shepp (a necessary and sufficient condition for random covering) and of the several dimensional situation (including E1 Helou's method). I describe Gaussian processes with stationary increments as helices in a Gaussian Hilbert space, and study the sample functions- mainly the sample functions of Brownian motion- in more detail. The last chapters show the interplay between Hausdorff dimensions, Lipschitz or Holder conditions, Gaussian processes, images, graphs, level sets, Fourier properties of measures. As a rough idea random sets obtained in this way fill the space as much as they can, and the spectra of random measures are smooth. Salem sets look very natural from this point of view.A few tools from probability theory
Random series in a Banach space
Random series in a Hilbert space
Random Taylor series
Random Fourier series
A bound for random trigonometric polynomials and applications
Conditions on coefficients for regularity
Conditions on coefficients for irregularity
Random point-masses on the circle
A few geometric notions
Random translates and covering
Gaussian variables and Gaussian series
Gaussian Taylor series
Gaussian Fourier series
Boundedness and continuity for Gaussian processes
The Brownian motion
Brownian images in harmonic analysis
Fractional Brownian images and level sets