Measures with Symmetry Properties

173 Pages·2003·1.33 MB·English

by  Werner Schindler (auth.)

1808
Lecture Notes in Mathematics
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Werner Schindler
Measures
with Symmetry Properties
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WernerSchindler
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derInformationstechnik(BSI)
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53175Bonn,Germany
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In memory of my father

Preface
Symmetry has played an important role in art and architecture since early
mankind. Symmetries occur in botany, biology, chemistry and physics. It is
therefore not surprising that in various branches of mathematics symmetries
and invariance principles have emerged and were put to good use as research
tools. A prominent example is measure and integration theory where a rich
theory has been developed for measures with remarkable symmetry proper-
ties such as Haar measures on locally compact groups, invariant measures
on homogeneous spaces, or particular measures on IRn with special symme-
tries. In connection with invariant statistics research has focussed on specific
situations in which measures that do not have ‘full’ symmetry occur.
Thisbookalsodealswithmeasureshavingsymmetryproperties.However,
our assumptions on the symmetry properties and on the general framework in
which we are working are rather mild. This has two somewhat contradictory
consequences. On the one hand, the weakness of the hypotheses we impose
and the ensuing generality cause the mathematics with which we have to
cope to be deep and hard. For instance, certain techniques which have been
very efficient in differential geometry are not available to us. On the other
hand, the generality of our approach allows us to apply it to an amazingly
wide range of problems. This new scope of applications is the positive side of
the generality of our attack.
The main results of this book belong to measure and integration the-
ory and thus to pure mathematics. However, in the second part of the book
many concrete applications are discussed in great detail. We consider inte-
gration problems, stochastic simulations and statistics. The examples span
from computational geometry, applied mathematics, computer aided graph-
ical design to coding theory. This book therefore does not only address pure
and applied mathematicians, but also computer scientists and engineers. As
a consequence, I have attempted to make the discourse accessible to readers
from a great variety of different backgrounds and to keep the prerequisites to
a minimum. Only in Chapter 2 where the core theorems are derived did it ap-
pear impossible to adhere strictly to this goal. However, I have arranged the
presentation in such a fashion that a reader who wishes to skip this chapter
completely, can read the remaining parts of the book without loss of under-
VIII Preface
standing. As far as they are relevant for the applications, the main results of
Chapter 2 are summarized at the beginning of Chapter 4.
This book arose from my postdoctoral thesis (Habilitationsschrift, [71]).
I would like to thank Karl Heinrich Hofmann, Ju¨rgen Lehn and Gunter Rit-
ter again for acting as referees and for their valuable advice, Siegfried Graf
for providing me with an interesting reference, and Hans Tzschach for the
encouragement he has given me to even consider the writing of this thesis.
Relative to the postdoctoral thesis, in the present book I have added some
new results and a number of illustrative examples. Of course there are ed-
itorial adjustments, and the text has been made more ‘self-contained’; this
should enhance its readability without forcing the reader to consult too many
references. I would like to thank Karl Heinrich Hofmann for his advice and
support. Finally, I want to thank my wife Susanne for her patience with me
when I was writing my postdoctoral thesis and this book.
Sinzig, Werner Schindler
October 2002
Table of Contents
1 Introduction .............................................. 1
2 Main Theorems........................................... 11
2.1 Definitions and Preparatory Lemmata..................... 11
2.2 Definition of Property (∗) and Its Implications (Main Results) 26
2.3 Supplementary Expositions and an Alternate Existence Proof 43
3 Significance, Applicability and Advantages................ 55
4 Applications .............................................. 63
4.1 Central Definitions, Theorems and Facts .................. 63
4.2 Equidistribution on the Grassmannian Manifold and
Chirotopes ............................................ 76
4.3 Conjugation-invariant Probability Measures on Compact
Connected Lie Groups .................................. 84
4.4 Conjugation-invariant Probability Measures on SO(n) ....... 90
4.5 Conjugation-invariant Probability Measures on SO(3) ....... 98
4.6 The Theorem of Iwasawa and Invariant Measures on Lie
Groups................................................ 115
4.7 QR-Decomposition on GL(n) ............................ 117
4.8 Polar Decomposition on GL(n)........................... 123
4.9 O(n)-invariant Borel measures on Pos(n) .................. 130
4.10 Biinvariant Borel Measures on GL(n) ..................... 134
4.11 Symmetries on Finite Spaces............................. 146
References.................................................... 155
Glossary ...................................................... 159
Index ......................................................... 165
1 Introduction
Symmetries and invariance principles play an important role in various
branches in mathematics. In measure theory and functional analysis Haar
measures on locally compact groups and invariant measures on homogeneous
spaces are of particular interest as they have a maximum degree of symmetry.
In particular, up to a multiplicative constant they are unique.
This book treats measures which are invariant under group actions.
Loosely speaking, we say that these measures have symmetry properties.
In general, these measures have a lower degree of symmetry than Haar mea-
sures or invariant measures on homogeneous spaces, and usually they are not
unique.
This work falls into two large blocks. Chapter 2 may be viewed as its
mathematical core as the main results are derived there. The main results
and their proofs mainly belong to different areas of measure and integration
theory. The importance of their own, their applicability and usefulness for
solving specific high-dimensional integrals, for stochastic simulations and in
statistics are explained and discussed in detail. In Chapter 4 these insights are
used to simplify calculations and to save computing time and memory. Par-
ticular problems become practically feasible only with these improvements.
We treat examples that range from Grassmannian manifolds and compact
connected Lie groups to various matrix groups and coding theory. Under
suitable conditions a priori bounds for the error propagation in numerical al-
gorithms for the computation of the polar decomposition of (n×n)-matrices
can be derived from a posteriori bounds. From this the necessary machine
precision can be determined before the first calculation rather than retrospec-
tively after the final one. Similarly, by exploiting symmetries a whole class of
random rotations in IRn can be simulated efficiently. Before we go into details
we illustrate and motivate the essential idea of our concept by a well-known
example.
For many computations it turns out to be profitable to use polar, spherical
or cylinder coordinates. In particular,
(cid:2) (cid:2) (cid:2)
∞ 2π
f(x,y)dxdy = rf (ϕ (α,r)) dαdr (1.1)
p
IR2 0 0
with ϕ (α,r) := (rcosα,rsinα). If f is radially symmetric, i.e. if the in-
p
tegrand f does only depend on the norm of its argument, the right-hand
W. Schindler: LNM 1808, pp. 1–9, 2003.
(cid:2)c Springer-Verlag Berlin Heidelberg 2003