This book presents in a concise and accessible way, as well as in a common setting, various tools and methods arising from spectral theory, ergodic theory and stochastic processes theory, which form the basis of and contribute interactively a great deal to the current research on almost everywhere convergence problems. The text is divided into four parts. Part I is devoted to spectral results such as von Neumann's theorem, spectral regularizations inequalities and their link with square functions and entropy numbers of ergodic averages. The representation of a weakly stationary process as Fourier transform of some random orthogonal measure, and a study of Gaposhkin's spectral criterion conclude this part. Classical results such as mixing in dynamical systems, Birkhoff's pointwise theorem, dominated ergodic theorems, oscillations functions of ergodic averages, transference principle, Wiener-Wintner theorem, Banach principle, continuity principle, Bourgain's entropy criteria, Burton-Denker's central limit theorem are covered in part II. The metric entropy method and the majorizing measure method, including a succinct study of Gaussian processes, are treated in part III, with applications to suprema of random polynomials. Part IV contains a |