The aim of this book is to give a rigorous and complete
treatment of various topics from harmonic analysis with a strong
emphasis on symplectic invariance properties, which are often
ignored or underestimated in the time-frequency literature. The
topics that are addressed include (but are not limited to) the
theory of the Wigner transform, the uncertainty principle (from the
point of view of symplectic topology), Weyl calculus and its
symplectic covariance, Shubin s global theory of
pseudo-differential operators, and Feichtinger s theory of
modulation spaces. Several applications to time-frequency analysis
and quantum mechanics are given, many of them concurrent with
ongoing research. For instance, a non-standard pseudo-differential
calculus on phase space where the main role is played by Bopp
operators (also called Landau operators in the literature) is
introduced and studied. This calculus is closely related to both
the Landau problem and to the deformation quantization theory of
Flato and Sternheimer, of which it gives a simple
pseudo-differential formulation where Feichtinger s modulation
spaces are key actors.
This book is primarily directed towards students or researchers
in harmonic analysis (in the broad sense) and towards mathematical
physicists working in quantum mechanics. It can also be read with
profit by researchers in time-frequency analysis, providing a
valuable complement to the existing literature on the topic.
A certain familiarity with Fourier analysis (in the broad sense)
and introductory functional analysis (e.g. the elementary theory of
distributions) is assumed. Otherwise, the book is largely
self-contained and includes an extensive list of references.
General
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