Recent Books
Integrating the Wigner Distribution on subsets of the phase space, a Survey,
Memoirs of the European Mathematical Society, EMS Press,
volume 12, (1), (2024), 224 pages,
link on HAL.
We review several properties of integrals of the Wigner distribution on subsets of the phase space. Along our way, we provide a theoretical proof of the invalidity of Flandrin's conjecture, a fact already proven via numerical arguments in our joint paper [MR4054880]
with B. Delourme and T. Duyckaerts. We use also the J.G. Wood & A.J. Bracken paper [MR2131219],
for which we offer a mathematical perspective.
We review thoroughly the case of subsets of the plane whose boundary is a conic curve and show that Mehler's formula can be helpful in the analysis of these cases, including for the higher dimensional case investigated in the paper
[MR2761287] by
E. Lieb and Y. Ostrover.
Using the Feichtinger algebra, we show that, generically in the Baire sense,
the Wigner distribution of a pulse in
L2(ℝn)
does not belong to
L1(ℝ2n)
providing as a byproduct a large class of examples of subsets of the phase space
ℝ2n on which the integral of the Wigner distribution is infinite.
We study as well the case of convex polygons of the plane, with a rather weak estimate depending on the number of vertices, but independent of the area of the polygon.
The files video-São Carlos and
video-Reims (in French) might serve as an introduction to the topic.
Carleman Inequalities: an Introduction and More,
a new book, published in 2019
in the Springer-Verlag Series
Grundlehren der Mathematischen Wissenschaften
.
This is a 576-page book, dealing with various aspects of Carleman inequalities.
On that
link, you will find a short description of the contents of the book as well as some previews opportunities.
A Course on Integration Theory, Including More Than 150 Exercises With Detailed Answers,
a book published by
Birkhäuser in 2014.
This volume is a 500-page textbook on Integration Theory, supplemented by 160
exercises
provided with detailed answers.
We have tried here to keep a rather elementary level,
at least in the way of exposing our arguments and proofs, which are
hopefully complete, detailed, sometimes at the cost of a lack of
concision.
Moreover, we hope that the many exercises (with answers) included at the
end of each chapter
will represent a key asset for the present book.
Here is
a pdf file of
Table of Contents and Preface of the Book.
Metrics on the Phase Space and Non-Selfadjoint Pseudodifferential Operators,
a book published by
Birkhäuser in 2010.
This is a four-hundred-page book on the topic
of pseudodifferential operators, with special emphasis
on non-selfadjoint operators, a priori estimates
and localization in the phase space.