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Diophantine Approximation and Dirichlet Series (Paperback)
Loot Price: R1,537
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Diophantine Approximation and Dirichlet Series (Paperback)
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This self-contained book will benefit beginners as well as
researchers. It is devoted to Diophantine approximation, the
analytic theory of Dirichlet series, and some connections between
these two domains, which often occur through the Kronecker
approximation theorem. Accordingly, the book is divided into seven
chapters, the first three of which present tools from commutative
harmonic analysis, including a sharp form of the uncertainty
principle, ergodic theory and Diophantine approximation to be used
in the sequel. A presentation of continued fraction expansions,
including the mixing property of the Gauss map, is given. Chapters
four and five present the general theory of Dirichlet series, with
classes of examples connected to continued fractions, the famous
Bohr point of view, and then the use of random Dirichlet series to
produce non-trivial extremal examples, including sharp forms of the
Bohnenblust-Hille theorem. Chapter six deals with Hardy-Dirichlet
spaces, which are new and useful Banach spaces of analytic
functions in a half-plane. Finally, chapter seven presents the
Bagchi-Voronin universality theorems, for the zeta function, and
r-tuples of L functions. The proofs, which mix hilbertian geometry,
complex and harmonic analysis, and ergodic theory, are a very good
illustration of the material studied earlier.
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