This book is a collection of articles
based on lectures given at an international conference on ‘Harmonic Analysis’
held at Delhi in the winter of 1995.
Most of the articles, though not all, centre around
the theme of harmonic analysis on hypergroups. While the term ‘harmonic
analysis’ needs no explanation, the expression ‘hypergroup’ may not be
familar to many people. Rather than launch into a technical definition of a hypergroup, we
will be content to say here that the double coset space of a non-compact connected
semi-simple Lie group modulo its maximal compact subgroup K, i.e. K\G/K
is perhaps one of the most important examples of such an object. The harmonic analysis on
this class of hypergroups is really the harmonic analysis of spherical functions on
semi-simple Lie groups. This again is a small but important part of the formidable area of
harmonic analysis of functions on a semi-simple Lie group, a field which was dominated by
the towering figure of Harish-Chandra during the period 1950–1980. (ref. 1). Given
the importance of just this single class of hypergroups, it is therefore perfectly
legitimate to study hypergroups in general. The theory of hypergroups was systematically
developed by Charles Dunkl, Robert Jewett and René Spector in the 70’s. It turns out
that hypergroups arise in a surprisingly large number of completely different situations.
The articles by K. A. Ross and A. L. Schwartz give a
comprehensive survey of the theory of hypergroups. Schwartz’s article ends with a
list of open problems. V. S. Sunder and N. J. Wildberger give a good introduction to the
theory of actions of finite hypergroups. (Readers interested in exploring the connections
between hypergroups and von Neumann algebras should consult ref. 2.) In ‘Wavelets on
Hypergroups’, K. Trimeche defines wavelets on hypergroups and proceeds to give an
account of the theory of continuous wavelet transforms on hypergroups. (In the context of
the real line, wavelet-theory happens to be one of the ‘hot’ areas of modern
Fourier analysis.)
Notable among the articles not directly connected
with the theme of hyper-groups are ‘Disintegration of Measures’ by H. Helson and
‘Multipliers of de Branges – Rovnyak spaces II’ by B. A. Lotto
and D. Sarason.
There are fifteen articles in all, but for the sake
of brevity we have restricted ourselves to mentioning just a small sample. This by no
means implies that the other articles are not interesting! To conclude, this collection of
articles will be very useful to researchers in harmonic analysis as well as advanced
Ph D students, and Ross et al. should be thanked for compiling this
collection.