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.
