This book deals with the theory of one- and two-parameter martingale
Hardy spaces and their use in Fourier analysis, and gives a summary of
the latest results in this field. A method that can be applied for both
one- and two-parameter cases, the so-called atomic decomposition method,
is improved and provides a new and common construction of the theory of
one- and two-parameter martingale Hardy spaces. A new proof of
Carleson's convergence result using martingale methods for Fourier
series is given with martingale methods. The book is accessible to
readers familiar with the fundamentals of probability theory and
analysis. It is intended for researchers and graduate students
interested in martingale theory, Fourier analysis and in the relation
between them.