Harmonic Analysis on the Heisenberg nilpotent Lie group.

206 Pages·1986·10.234 MB·English

by  W. Schempp

WSchempp
University of Siegen
Harmonic analysis
on the Heisenberg
nilpotent Lie group,
with applications
to signal theory
...,. • ...,. Longman
• • • Scientific &
•..•.•. ~.i ech n1. caI
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John Wiley & Sons, Inc., New York
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©
W Schempp 1986
All rights reserved; no part of this p1 iblication
may be reproduced, stored in a retril!val system,
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without the prior written permission of the Publishers.
First published 1986
AMS Subject Classifications: (main I 22E27, 43A35, 94A12
(subsidiary) 22010, 22E25, 41A15
ISSN 0269-3674
British Library Cataloguing in Publication Data
Schempp. W.
Harmonic analysis on the Heisenl •erg
nilpotent Lie group, with applicat ons to
signal theory .-(Pitman research 1otes
in mathematics, ISSN 0269-3674; 147)
1. Lie groups, Nilpotent
I. Title
512' .55 QA387
ISBN 0-582-99453-5
Library of Congress Cataloging-in-l'ublication Data
Schempp. W. (Walter), 1938-
Harmonic analysis on the Heisenl •erg nilpotent Lie
group, with applications to signal th !ory.
(Pitman research notes in mathen 1atics series,
ISSN 0269-3674; 147)
Bibliography: p.
Includes index.
1. Harmonic analysis. 2. Lie grou.>s, Nilpotent.
3. Signal theory (Telecommunicati(ln) I. Title.
II. Title: Heisenberg nilpotent Lie~ roup, with
applications to signal theory. III. Se des: Pitman
research notes in mathematics; 147.
QA403.S27 1986 515' .2433 8 >-13233
1
ISBN 0-470-20374-9 (USA only)
Printed and bound in Great Britain l>y
Biddies Ltd, Guildford and King's l ynn
Contents
Pref ace
0. Basic notations and conventions
1. Basic facts on linear group reprt·sentations 2
2. The unitary inducing procedure 32
3. Square integrable linear group rr·presentat ions 58
4. Basic facts en real nilpotent Lie groups 75
5. The real Heisenberg nilpotent Lit group. Part I 101
6. The coadjoint orbit picture 119
7. The real Heisenberg nilpotent LiP group. Part II 140
8. Applications to signal theory 168
Index 197
Prefa ce
The real Heisenberg group A(R) is connected and simply_connected,
:a tw~-step
nilpotent, analytic group having 0·1e-dimensional centre C. Therefore A(R)
fonns the simplest possible non-contnutative, non-compact Lie group. The name
and the quantum mechanical meaning of the real Hei-senberg nilpotent Lie group
...
A(R) stem from the fact that the Lie algebra n of A(R )over R is defined by
the Heisenberg canonical co1T111uta ti ·:>n re 1a t ions. Thus , according to the
philosophy of Niels Bohr, the intuition necessarily fails to
geom·~tric
describe the action of A(R). It i '; the purpose of these notes to study nil
potent harmonic anal-ysis in a unified manner and specifically to determine
the unitary dual of A(R) by an application of the Mackey machinery as well as
by the Kirillov orbit picture. coadjoint orbit method provides a deep
Th>~
geometric insight into the har-moni.: analysis of the Heisenberg Lie group.
Although the unitary dual of A(R) is extremely poor, there are many rather
different looking ways of realizin1 the non-degenerate, topologically irre-
-
ducible, continuous, unitary, lineir representations of A(R). This important
fact ad-ds greatly to the applicability of the real Heisenberg nilpotent Lie
group A(R) and turns it into a far reaching tool for various different areas
of pure and applied mathematics, t·1eoretical physics, information theory, and
electrical engineering. In the notes, however, the main emphasis of
pr:~sent
the applications are laid on the t'1eory of analog and digital signals since
the group theoretical ideas behind this subject have been discovered quite
recent 1y . Spec i fi ca 11 y the notes present the so 1u t ions of two prob 1e ms of
analog radar signal design: the synthesis problem of characterizing intrins
ically the bivariate analog radar Jute-ambiguity functions and the invariant
problem of computational signal geJmetry of calculating explicitly the linear
energy preserving automorphisms of the radar ambiguity surfaces over the
symplectic time-frequency plane. fJoth the solutions are achieved via harmonic
analysis on the differential principal fibre bundle over the two-dimensional
polarized resp. isotropic cross-section with structure group isomorphic to the
C
centre of A(R). Moreover it it shown how the linear lattice repr~sentation
...
of A(R} gives rise to a geometric proof of the sampling theorem of digital
signal processing and how to deduce basically from the precedi.ng results some
new identities for Laguerre functions and theta-null values.
Those parts of the notes which are concerned with elementary group repre
sentation theory are based on lectur'es entitled EinfUhrung in die Darstell
11
ungstheorie lokalkompakter topologi';cher Gruppen given by the author at the
11
University of Siegen in Winter Seme,;ter 1983/84. Apart from the theory of
analog and digital signals there ar,?, however, various other applications of
harmonic analysis on the principal Jifferential fibre bundle over the two
dimensional polarized resp. isotropic cross-section with structure group
C
isomorphic to the centre of A(R), and the closely related theory of the
Segal-Shale-Weil metaplectic (or li lear oscillator) representation for reduc
tive dual pairs in metaplectic grou>s. Among the applications which are of
actual interest from the technologi :al point of view we should particularly
emphasize the field of beam optics, the synthesis for dielectric multilayer
filters, the theory of transmission by dielectric waveguides and optical
distributed-index round fibres, the design of maser and laser resonators
including optical phase conjugation devices, and the holographic imaging.
Unfortunately a detailed treatment ,,f these topics and their applications to
optical communication systems of hi 1h capacity are outside the scope of the
present notes. However these notes will form the foundation of a research
program dealing with a group repres1·ntational approach to certain optical
phenomena. A series of invited pap1·rs appearing in the near future will
trace a research line which follows Charles H. Townes' example by starting
from analog radar signal design and leading via group theoretical arguments
to a detailed treatment of various cevices of microwave and laser optics.
Walter Schempp
Lehrstuhl fuer Mathematik I
University of Siegen
Acknowledgements
First and foremost the author wishf's to acknowledge the invaluable stimulation
and encouragement supplied by Aline Bonami (Orleans), Chen Han-lin
Proft~ssors
(Beijing), Charles K. Chui (Collegt· Station), Lothar Collatz (Hamburg),
Phillipe Combe (Marseille), Rudolf de Buda (Toronto), Walter Gautschi (West
Lafayette), Mi chi el Hazewinkel (Am'.·.terdam), Edwin Hewitt (Seattle), J. Rowland
Higgins (Cambridge), Hu Ying-sheng (Beijing), Mourad E.H. Ismail (Tempe),
Palle E.T. Jr6rgensen (Iowa City), ,!ohn R. Klauder (Murray Hill), Adam
Kor~nyi
( Bronx ) , Peter Kramer (TUbi nge1 ) , Giovanni Monegato (Tori no), GUnter Ries
(Siegen), Rudolf Schwarte (Siegen), Harvey A. Smith (Tempe), Orestes N.
Stavroudis (Tucson), Niels tetkaer (Arhus), Daniele C. Struppa (Pisa),
Henrik~
Kurt Bernardo Wolf (Mexico City), .'nd the microwave engineer Jerzy Brzeski
(Wa 1n ut Creek).
Moreover, the author wishes to the various Universities, Research
~hank
Institutes, and funding agencies w1 ii ch have offered research faci 1i ti es,
hos pi ta 1i ty, and support during th1 conduct of the research parts of which
~
are reported here. Speci fi ca lly, author s thanks go to Aarhus Uni vers i tet
~he 1
at A0 rhus, Denmark; Academia Sinica at Beijing, The People's Republic of China;
Arizona State University at Tempe, Arizona; the Bulgarian Academy of Sciences
at Sofia, Bulgaria; the Center for Approximation Theory at College Station,
Texas; to Centro de Investigacione, en Optica at Leon, Guanajuato, Mexico; to
the Centrum voor Wiskunde en Inforlllatica at Amsterdam, The Netherlands; to
the Mathematisches Forschungsinstitut Oberwolfach, Black Forest, Germany; to
the Scuola Normale Superiore at Pi ;a, Italy; to the University of Alaska at
Fairbanks, Alaska; University of at Newark, Delaware; University of
D1~laware
Illinois, Urbana-Champaign, Illinois; University of Maryland, College Park,
Maryland; Universidad Nacional Aut.jnoma de at Mexico City: Universite
M~xico
d'Orleans at Orleans, France; Penn;ylvania State University at University
Park, Pennsylvania; Seoul National University in Seoul, Korea; di
Universi~a
Torino in Turin, Italy; University of Washington at Seattle, Washington,
Zentrum fuer interdisziplinaere For·schung (ZiF) at Bielefeld, Germany;
Consiglio Nazionale delle Ricerche of Italy, Deutscher Akademischer
Austauschdienst, Deutsche Forschun1sgemeinschaft, Korean Physical Society,
1
Scuola Matematica Interuniversitaria, and finally, to the National Science
Foundation.
0 Basic notations !tnd conventions
First, the symbols TI, Z, R, and [denote the sets of natural, integral,
~'
rational, real, and complex number~, respectively: Rx denotes the set of real
numbers # 0 and T the set of comp l numbers of modu 1u s 1 • Throughout these
'~X
notes we shall adhere to the follovJing conventions. By the terms 'locally
compact to po 1o gi ca 1 group' and group we shall understand a 1oca11 y
'Li ·~ 1
compact topological group and a Li? group that are countable at infinity. A
simply connected Lie group shall a connected and simply connected Lie
m~an
group. Finally, by the term 'Hil b·?rt space' we sha 11 mean a separab 1e Hil ber
space.
At the end of each chapter is a list of references which supply
ther·~
additional material. Special emphisis is laid on survey articles which
include further references.
1 Basic facts on linear group representations
1.1 Let G denote a group. Write its group law in the multiplicative way
and denote by 16 the neutral element of G. A linear representation of the
group G in a complex vector space 11 is a pair (U,H) where U denotes a mapping
which assigns to every element x E Ga t-linear mapping U(x) : H-.. It such
that the following two conditions are satisfied:
(I) U(1 ) = idH (the identity operator of H);
6
(II) For all pairs (x,y) E G x G \':e have U(xy) = U(x) 0 U(y).
Obviously we have U(x) U(x-1) = idH and therefore U(x-1) = (U(x))-1 for all
o
x € G. Thus a linear representation (U,H) of G in the representation spaae It
defines a morphism x/\/'? U(x) of U:e group G into the group Aut(H) = ~k(H) of
automorphisms of H. In the case when G is a topological group and H denotes
a topological vector space over th£· field t, a linear representation (U,H) of
G is said to be aontinuous if the linear left G-action on H canonically defined
by the assignment
G H (x,f)IV'? U(x)f H
x 3 €
is a continuous mapping with respect to the product topology of G H and the
x
given vector space topology of IC. Thus the left G-module H becomes a topo
logiaal left G-module. More specifically, when (H;<.!.>) is a complex Hilbert
space with norm II· II associated \-Jith its scalar product <.1.>, a linear
representation (U,H) of G is said to be unitary if the automorphism U(x) € ~k(H)
forms a unitary operator of ff for all x E G. In this case U defines a morphism
x~ U(x) of the group G into the unitary group M(H) of ff such that U(x-1) =
U(x)* holds for all x E G. Clearly the unitarity of the linear representation
(U,H) of G is equivalent to one of the following two equivalent conditions:
(i) llU(x)fll=llfll for all x E G and all f EH;
(ii) <U(x)f!U(x)g> =<fig> for all x € G and all pairs (f,g) EH x H.
2
In the case when (U,H) is a unitary linear representation of the topological
group G in the complex Hilbert space JC the property of being continuous as
defined previously follows if (U,H) is merely supposed to be separately
continuous, to wit, if the mapping given by
G 3 x U(x)f E H
~
is supposed to be continuous for any choice of the vector f E H. Indeed,
let x E G and f E H be fixed elen·ents. For any given £ > 0 there exists
0 0 1
a suitable neighbourhood V of x in G such that
0 0
holds for all x E V • Let the vector f EH satisfy llf - f ll < ~£. By
0 0
virtue of (i) we get for all elements x E V the estimate
0
1 1
< "2'£ + "ZE: = £.
Thus the linear left G-action (x,f: U(x)f associated with (U,H) is con
~
tinuous in all points (x ,f E G Hand hence globally continuous on the
) >
0 0
topological product space G x H. The topology induced be the strong operator
topology of the complex vector space End(H) of aontinuous endomorphism of H
on the unitary group M(H) coincides. with topology induced by the weak operator
topology of End(H) on M(H) and is c-ompatibl,e with the group structure of M(H).
1.2 Theorem. A unitary linear representation (U,H) of the topological group
Gin the complex Hilbert space H is continuous if and only if the morphism of
topological groups
G 3 x ~ U(x) E ~(H)
is continuous.
The topology on End(H), i .e·., the topology of uniform convergence on
norm
the bounded subsets of H, is finer than the strong operator topology on
3