COHERENT STATES, WAVELETS, AND THEIR GENERALIZATIONS

Should authors feel compelled to justify the writing of yet another book? In an overpopulated
world, must parents feel compelled to justify the bringing forth of yet another child?
Perhaps not! But an act of creation is also an act of love and a love story can always be happily
shared.
In writing this book, it has been the authors’ feeling, that in all the wealth of material on
coherent states and wavelets, in which the literature on the subject abounds, there is lack of a
discernable unifying mathematical perspective. The use of wavelets in research and technology
has witnessed explosive growth in recent years, while the use of coherent states in numerous
areas of theoretical and experimental physics has been an established trend since decades. Yet
it is not at all uncommon to nd practitioners in either one of the two disciplines, who are
hardly aware of its links to the other. Currently, there are many books in the market which
treat the subject of wavelets from a wide range of perspectives and with windows on one or
several areas of a large spectrum of possible applications. On the theory of coherent states,
likewise, there exist several excellent monographs, edited collections of papers, synthetic reviews
or specialized articles. The emphasis in most of these works is usually on physical applications.
In the more mathematical works, the focus is usually on specic properties { arising either
from group theory, holomorphic function theory and, more recently, dierential geometry. The
point of view put forward in this book, is that both the theory of wavelets and the theory
of coherent states can be subsumed into certain broad functional analytic structures, namely,
positive operator valued measures on a Hilbert space and reproducing kernel Hilbert spaces. The
specic context in which these structures arise, to generate particular families of coherent states
or wavelet transforms, could of course be very diverse, but typically they emanate either from
the property of square integrability of certain unitary group representations on Hilbert spaces,
or from holomorphic structures associated to certain dierential manifolds. In talking about
square integrable representations, a broad generalization of the concept has been introduced
1Thine is an eternal celebration … { A cosmic Festival of Lights! … Therein I am a mere
here, moving from the well known notion of square integrability with respect to the whole group,
to one based upon some of its homogeneous spaces. This generalization, while often implicit in
the past, in physical discussions of coherent states (notably, in the works of Klauder, Barut and
Girardello, or even in the case of the time honoured canonical coherent states, discovered by
Schrodinger), had not been readily recognized in the mathematical literature until the work of
Gilmore and Perelomov. In this book, this generalization is taken even further, with the result
that the classes of coherent states that can be constructed and usefully employed extends to a vast
array of physically pertinent groups. Similarly, it is generally known and recognized that wavelets
are coherent states arising from the ane group of the real line. But using coherent states of
other groups to generate higher dimensional wavelets, or alternative wavelet-like transforms, is
not such a common preoccupation among practitioners of the trade (an exception being the
recent book by Torresani). About a third of the present book is devoted to looking at wavelets
from precisely this point of view, displaying thereby the richness of possibilities that exists in this
domain. Considerable attention has also been paid in the book to the discretization problem,
in particular with the discussion of -wavelets. The interplay between discrete and continuous
wavelets is a rich aspect of the theory, which does not seem to have been exploited suciently
in the past.
In presenting this unifying backdrop, for the understanding of a wide sweep of mathematical
and physical structures, it is the authors’ hope that the relationship between the two disciplines
{ of wavelets and coherent states { will have been made more transparent, aiding thereby the
process of cross-fertilization as well. For graduate students or research workers, approaching the
disciplines for the rst time, such an overall perspective should also make the subject matter
easier to assimilate with the book acting as a dovetailed introduction to both subjects { unfortunately,
a frustrating incoherence blurs the existing literature on coherence! Besides being
a primer for instruction, the book, of course, is also meant to be a source material for a wide
range of very recent results, both in the theory of wavelets and of coherent states. The emphasis
is decidedly on the mathematical aspect of the theory, although enough physical examples have
been introduced, from time to time, to illustrate the material. While the book is aimed mainly
at graduate students and entering research workers in physics and mathematical physics, it is
nevertheless hoped that professional physicists and mathematicians would also nd it interesting
reading, being an area of mathematical physics in which the intermingling of theory and practice
is most thoroughgoing. Prerequisites for an understanding of most of the material in the book is
a familiarity with standard Hilbert space operator techniques and group representation theory,
such as every physicist would acknowlegde from the days of graduate quantum mechanics and
angular momentum. For the more specialized topics, an attempt has been made to make the
treatment self-contained and indeed, a large part of the book is devoted to the development of
the mathematical formalism.
If a book such as this can make any claims to originality at all, it can mainly be in the
manner of its presentation. Beyond that, the authors believe that there is also a body of material
presented here (for example in the use of POV measures or in dealing with the discretization
problem), which has not appeared in book form before. An attempt has been made throughout
to cite as many references to the original literature as were known to the authors { omissions
should therefore be attributed to their collective ignorance and the authors would like to extend
their unconditional apologies for any resulting oversight.
This book has grown out of many years of shared research interest and indeed, camaraderie,
between the three authors. Almost all of the material presented here has been touched upon in
courses, lectures and seminars, given to students and among colleagues at various institutions
in Europe, America, Asia and Africa { notably in graduate courses and research workshops,
given at dierent times by all three authors, in Louvain-la-Neuve, Montreal, Paris, Porto-Novo,
Bia lystok, Dhaka, Fukuoka, Havana and Prague. One is tempted to say that the geographical
diversity here rivals the mathematical menagerie!
To all their colleagues and students who have participated in these discussions, the authors
would like to extend their heartfelt thanks. In particular, a few colleagues graciously volunteered
to critically read parts of the manuscript and to oer numerous suggestions for improvement and
clarity. Among them, one ought to specially mention J. Hilgert, G. G. Emch, S. De Bievre and
J. Renaud. In addition, the gures would not exist without the programming skills of A. Coron,
L. Jacques and P. Vandergheynst (Louvain-la-Neuve), and we thank them all for their gracious
help. During the writing of the book, the authors made numerous reciprocal visits to each others’
institutions. To Concordia University, Montreal, the Universite Catholique de Louvain, Louvainla-
Neuve and the Universite Paris 7 { Denis Diderot, Paris, the authors would like to express
their appreciation for hospitality and collegiality. The editor from Springer-Verlag, Thomas von
Foerster, deserves a special vote of thanks for his cooperation and for the exemplary patience he
displayed, even as the event horizon for the completion of the manuscript kept receeding further
and further! It goes without saying, however, that all responsibility for errors, imperfections
and residual or outright mistakes, is shared jointly by all three authors.