the attention of The publication of Charles Pisot's thesis in 1938 brought to the mathematical community those marvelous numbers now known as the Pisot numbers (or the Pisot-Vijayaraghavan numbers). Although these numbers had been discovered earlier by A. Thue and then by G. H. Hardy, it was Pisot's result in that paper of 1938 that provided the link to harmonic analysis, as discovered by Raphael Salem and described in a series of papers in the 1940s. In one of these papers, Salem introduced the related class of numbers, now universally known as the Salem numbers. These two sets of algebraic numbers are distinguished by some striking arith metic properties that account for their appearance in many diverse areas of mathematics: harmonic analysis, ergodic theory, dynamical systems and alge braic groups. Until now, the best known and most accessible introduction to these num bers has been the beautiful little monograph of Salem, Algebraic Numbers and Fourier Analysis, first published in 1963. Since the publication of Salem's book, however, there has been much progress in the study of these numbers. Pisot had long expressed the desire to publish an up-to-date account of this work, but his death in 1984 left this task unfulfilled.
The second edition of the book includes a new chapter on the study of composition operators on the Hardy space and their complete characterization by Gordon and Hedenmalm. The book is devoted to Diophantine approximation, the analytic theory of Dirichlet series and their composition operators, and connections between these two domains which often occur through the Kronecker approximation theorem and the Bohr lift. The book initially discusses Harmonic analysis, including a sharp form of the uncertainty principle, Ergodic theory and Diophantine approximation, basics on continued fractions expansions, and the mixing property of the Gauss map and goes on to present the general theory of Dirichlet series with classes of examples connected to continued fractions, Bohr lift, sharp forms of the Bohnenblust–Hille theorem, Hardy–Dirichlet spaces, composition operators of the Hardy–Dirichlet space, and much more. Proofs throughout the book mix Hilbertian geometry, complex and harmonic analysis, number theory, and ergodic theory, featuring the richness of analytic theory of Dirichlet series. This self-contained book benefits beginners as well as researchers.
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