I1: r162 -

- ,

[~kVA 1 . ~ 1

Jet Wimp,

Editor

/

Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture by David M. Bressoud CAMBRIDGE: CAMBRIDGEUNIVERSITYPRESS, 1999,

Ank=( X

k-k-2) (2n-k1)! (n - k)!

~:r2 (32"+1)! try=0 (n +3)!

and therefore that

An =

1 ( 3 j + 1)! j=0 (n + j ) ! "

xv + 274 pp. HARDCOVER, US $74.95, ISBN 0 521 66170 6; SOFTCOVER, US $29.95 ISBN 0 521 66646 5.

REVIEWED BY MARTIN ERICKSON

Feel like writing a review for The Mathematical Intelligencer? You are welcome to submit an unsolicited review of a book of your choice; or, if you would welcome being assigned a book to review, please write us, telling us your expertise and your predilections.

n alternating sign matrix is a A square array of 0s, ls and - l s such that the non-zero entries of each row and of each column alternate in sign, and each row sum and column sum is 1. These matrices generalize permutation matrices. Here are the seven alternating sign matrices of order three:

00010 (!

!1) 0 0

01 00) 10

( 0i ) OO0 11

(i

(~0~) 0

(00~) 0 1

1

1 0

i ) 1- l l

.

In the early 1980s, mathematicians William Mills, David Robbins, and Howard Rumsey wondered whether they could find a formula for A n , the number of alternating sign matrices of order n. Notice that the definition forces each of the "borders" (first and last columns and rows) of an alternating sign matrix to consist of a single 1 and all other entries 0. Letting An,k be the number of alternating sign matrices of order n in which the first row's 1 occurs in column k, we have A n = A n + l , 1 --

Column Editor's address: Department of Mathematics, Drexel UniVersity, Philadelphia, PA 19104 USA.

An+l,n+l.

Based on knowledge of An,k for the first twenty values of n, the three researchers conjectured that

The validity of the latter formula is known as the Alternating Sign Matrix Conjecture. David M. Bressoud's book is the story of ~ conjecture, culminating in its proof in 1995 by Doron Zeilberger. Along the way, the author searches out and explicates the connections between alternating sign matrices and a host of other combinatorial topics, including generating functions, partitions, determinants, lattice paths, inversion numbers, plane partitions, symmetric functions, Schur fimctions, Young tableaux, hypergeometric series, and square ice (a model of H20 molecules frozen in a square lattice). The story also touches on the lives and contributions of many great mathematicians of the past, including Leibnitz, Euler, Lagrange, Gauss, Waring, Cauchy, Jacobi, Boole, Sylvester, and Ramanujan, as well as contemporary researchers such as Ian Macdonald, Richard Stanley, Donald Knuth, George Andrews, John Stembridge, and Greg Kuperberg (who discovered a different proof of the conjecture in 1995). Even Lewis Carroll makes an appearance (via Dodgson's algorithm). In the author's conception, the process by which the conjecture was investigated and eventually proved illustrates a way of looking at mathematics that differs from the standard paradigms. It is not theorem-proof-corollary; it is not "scaling the peaks" (moving from accomplishment to greater accomplishment); and it is not random exploration. By analogy with archaeology, Bressoud suggests a new way of de* scribing what mathematicians do:

9 2000 SPRINGER-VERLAGNEWYORK,VOLUME22, NUMBER4, 2000

71

"I w o u l d like to c o n s i d e r the doing o f m a t h e m a t i c s a n d the fmding of p r o o f s as analogous t o t h e w o r k of the archaeologist. When Mills, Robbins, and R u m s e y first d i s c o v e r e d their conjecture, t h e y w e r e n o t dissimilar to the archaeologist w h o h a s j u s t u n e a r t h e d a strange and m a r v e l o u s object o f unk n o w n p r o v e n a n c e a n d purpose. What is it? What w a s it u s e d for? Why is it here? What does it tell us a b o u t the p e o p l e w h o once lived here? The real w o r k o f the a r c h a e o l o g i s t is to m a k e connections: c o n n e c t i o n s to o t h e r obj e c t s at o t h e r places, at other times, c o n n e c t i o n s to o t h e r facts that are k n o w n a b o u t this p a r t i c u l a r site. The goal of the a r c h a e o l o g i s t is to p r o v i d e a c o n t e x t in which w e c a n u n d e r s t a n d this object. As e a c h o b j e c t c o m e s to be understood, it facilitates the interpretation o f others, n o t j u s t in this place, b u t also in o t h e r p l a c e s a n d from o t h e r times. It p r o v i d e s a f o u n d a t i o n u p o n which w e c o n s t r u c t o u r theories. "This is the role o f proof, to enrich the entire w e b o f c o n t e x t that l e a d s to understanding. The m a t h e m a t i c i a n d o e s not dig for lost artifacts of a vanished civilization b u t for the fundam e n t a l p a t t e r n s that u n d e r g r i d our universe, a n d like the a r c h a e o l o g i s t w e usually find only small fragments. As a r c h a e o l o g y a t t e m p t s to r e c o n s t r u c t the society in w h i c h this o b j e c t w a s used, so m a t h e m a t i c s is the reconstruction o f t h e s e p a t t e r n s into t e r m s that we can c o m p r e h e n d . : I believe that B r e s s o u d successfully illustrates his thesis in this book. The details are not a l w a y s e a s y to follow, as there are m a n y r e l a t e d c o n j e c t u r e s and m u c h m a t h e m a t i c a l m a c h i n e r y to k e e p t r a c k of. (Most o f t h e c o n j e c t u r e s have b e e n proved, b u t the a u t h o r refers to t h e m as conjectures, r a t h e r than theorems, in o r d e r to p r e s e r v e the historical p o i n t of view.) But the aut h o r d o e s a g o o d j o b o f helping us k e e p the main p o i n t s in mind, and his narration is quite clear. This is enjoyable h i s t o r y of m o d e r n m a t h e m a t i c s of the t y p e found in the b o o k From Error-Correcting Codes

Through Sphere Packings to Simple Groups, b y T h o m a s Thompson. As in that book, w e are t r e a t e d to quotes from the investigators, photographs,

72

THE MATHEMATICALINTELLIGENCER

lots o f diagrams, and m u c h backg r o u n d information. In fact, t h e main s t o r y o f the Alternating Sign Matrix C o n j e c t u r e is told in C h a p t e r s 1 and 6, while Chapters 2 through 5 a n d 7 provide r e l a t e d material. The a u t h o r also s u p p l i e s Mathematica code so that the r e a d e r can obtain d a t a and follow the d i s c u s s i o n in an active way. This is an e x c e l l e n t idea, a n d t h e c o d e is simple e n o u g h to copy a n d u s e in a few minutes. I e n j o y e d doing s o immensely. Proofs and Confirmations is a fascinating look at m a t h e m a t i c s in the making. And it is a g e n e r o u s guide to a d d i t i o n a l information that is interesting in its o w n right.

Department of Mathematics and Computer Science Truman State University Kirksville, MO 63501 USA e-mail: [email protected]

Physics from Fisher Information: A Unification by B. Roy Frieden CAMBRIDGE: CAMBRIDGEUNIVERSITYPRESS (1999), 318 pp. US $74.95, ISBN: 052163167X

REVIEWED BY ROBERT GILMORE

t is h a r d not to b e s e d u c e d b y a b o o k w h o s e very first equation is the Cramer-Rao inequality. This is a pror o u n d l y beautiful a n d i m p o r t a n t result w h i c h m a y be written p o e t i c a l l y as

I

A y A a _> 1. It is beautiful b e c a u s e it is simple. It is i m p o r t a n t b e c a u s e it is far-reaching: it manifests itself in m a n y different w a y s in m a n y b r a n c h e s o f physics. It is k n o w n to s o u n d e n g i n e e r s in the form o f a time-frequency u n c e r t a i n t y relation 1 A~oAt _ - 2' w h e r e At is the time d u r a t i o n o f a signal a n d A o = 27rA~ is the p r e c i s i o n to

which the a n g u l a r frequency can b e estimated. It o c c u r s in Quantum Mechanics as b o t h the p o s i t i o n ( x ) - m o m e n t u m (p) u n c e r t a i n t y relation h x A p ___lh and the t i m e - e n e r g y uncertainty relation 1 &EAt _> ~ h , w h e r e h is P l a n c k ' s c o n s t a n t and h = h/2cr. It a p p e a r s also in Statistical Mechanics in t h e form o f u n c e r t a i n t y relations b e t w e e n extensive variables (U,V,N, ...) (internal energy, volume, n u m b e r o f particles, ...) and their conj u g a t e intensive variables (1, P, -T_a, ...) (temperature, pressure, c h e m i c a l potential, 9..), o n e o f which is

w h e r e k is Boltzmann's constant. The Cramer-Rao inequality exhibits the duality w h i c h exists b e t w e e n t h e t w o fields: Probability Theory a n d Statistics. Specifically, a s s u m e t h a t P(y]o0 is a p r o b a b i l i t y distribution function for t h e r a n d o m variable Y. This p r o b a b i l i t y distribution d e p e n d s on the values o f one o r m o r e p a r a m e ters ~. Then, given c~, it is p o s s i b l e to estimate the various statistics o f t h e r a n d o m variable E such as its m e a n ?~ = (y) a n d s t a n d a r d deviation Ay, w h e r e Ay 2 = ((y - ?))2). Conversely, it is p o s s i b l e to e s t i m a t e the value o f t h e p a r a m e t e r a from the m e a s u r e m e n t s Yi. These e s t i m a t e s lead to a m e a n value &, a n d a s t a n d a r d deviation As, defined b y A a 2 = ( ( a - &)2). The p r o d uct o f t h e s e t w o variances, of the rand o m variable a n d the p a r a m e t e r estimates, are b o u n d e d b e l o w b y a nonnegative t e r m which can often b e n o r m a l i z e d to + 1 b y suitable change in the defmition o f the r a n d o m variable and/or the p a r a m e t e r s . The t e r m Ay 2 is called the F i s h e r information Ay 2 = I(oz) :

'

) rtyla)ay.

F i s h e r i n f o r m a t i o n is the starting point o f t h e j o u r n e y on w h i c h the aut h o r of the p r e s e n t b o o k e m b a r k s .

F r i e d e n has certainly m a d e a n u m b e r of useful observations: 1) Uncertainty relations p l a y a fund a m e n t a l role in m o d e m physics. 2) F i s h e r information, t h r o u g h the Cramer-Rao inequalities, u n d e r l i e s all u n c e r t a i n t y relations. 3) The dynamical laws o f p h y s i c s can all be f o r m u l a t e d as variational principles. These involve integrals over q u a d r a t i c forms in the g r a d i e n t s of suitable functions. 4) F i s h e r information has a similar s t r u c t u r a l form. On the basis o f these observations, F r i e d e n a t t e m p t s to create a vision of p h y s i c s as a s t e p c h i l d of I n f o r m a t i o n Theory, specifically of F i s h e r information. His thesis is that " . . . all p h y s i c a l law, from the Dirac equation to the Maxwell-Boltzmann velocity dispersion law, m a y be unified u n d e r the umb r e l l a o f classical m e a s u r e m e n t theory. In particular, the information a s p e c t of m e a s u r e m e n t t h e o r y - - F i s h e r I n f o r m a t i o n - - i s the k e y to unification." F r i e d e n ' s efforts have c r e a t e d a c o n s i d e r a b l e buzz in the w o r l d o f scie n c e a r e c e n t l e a d article in the journal The New Scientist a l l u d e d v e r y fav o r a b l y to F r i e d e n ' s book. However, s p e a k i n g as a card-carrying physicist w h o h a s derived the u n c e r t a i n t y relations for Statistical Mechanics using the Cramer-Rao inequality, I d o n o t believe t h a t F r i e d e n has s u c c e e d e d in his effort. The variational formulation o f the laws o f d y n a m i c s is a very p o w e r f u l tool in the physicists' bag o f tricks. The m e t h o d itself d a t e s b a c k to t h e 17th century, to F e r m a t ' s Principle o f Least Time, a n d to the 18th century, in the form o f Maupertuis's Principle o f Least Action. The original f o r m u l a t i o n s had a distinctly theological formulation. Even today, the m e t h o d smells faintly sulfurous. Here I will give a w a y one o f the t r i c k s of o u r trade. We p u t e x a c t l y "the right stuff" into a Lagrangian, that is, w h a t e v e r is n e c e s s a r y to r e c o v e r the d e s i r e d d y n a m i c s once the b u t t o n is p u s h e d and the m a c h i n e r y g o e s into gear. The m a c h i n e r y involves a Taylor e x p a n s i o n a r o u n d the a p p r o p r i a t e solution, one or m o r e integrations b y parts, a n d then an a r g u m e n t a b o u t an integrand n e c e s s a r i l y being zero. The

final result is s o m e equation o f statics o r dynamics. What m a k e s the variational m e t h o d so attractive is that there are c e r t a i n rules which severely restrict the form o f the Lagrangian function. This drastically r e d u c e s the g u e s s w o r k required in deriving dynamical laws to d e s c r i b e n e w physics. F r i e d e n e n c o u n t e r s three p r o b l e m s in trying to formulate physics from F i s h e r information. I illustrate t h e m for a p a r t i c u l a r l y simple case: the quant u m - m e c h a n i c a l d e s c r i p t i o n of a particle o f m a s s m moving in one dimension in a p o t e n t i a l V(x). The variational p r o b l e m is

0x

[ 2m \ Ox ] + V(x)(~b(x)) 2 dx = O. F o r simplicity, I have taken the wave function to b e real. In the p r o b l e m as specified, t h e r e is always the trivial solution ~b(x) = 0. To eliminate the trivial solution it is useful to i m p o s e the normalization constraint f ( 0 ( x ) ) 2 d x = 1. This c o n s t r a i n t is n o r m a l l y i m p o s e d through the u s e of Lagrange multipliers, so that t h e c o n s t r a i n e d variational problem becomes

8 \ ~ L--2-m-m o x J

]

The F i s h e r I n f o r m a t i o n is

I(o0 = ( ( OPI2

1

At this p o i n t in the analysis w e are faced with t h r e e p r o c e d u r a l p r o b l e m s : 1) The g r a d i e n t s are with r e s p e c t to the spatial c o o r d i n a t e s a n d the par a m e t e r s o f a p r o b a b i l i t y distribution function, respectively. 2) The p h y s i c a l Lagrangian has a potential term, V(x)(tp(x)) 2, a n d a constraint term, - h ( 0 ( x ) ) 2 ; the F i s h e r inf o r m a t i o n h a s neither. 3) The p h y s i c a l law involves a variation, the F i s h e r information d o e s not. F r i e d e n a t t e m p t s to solve the fn'st p r o b l e m b y a s s u m i n g the p r o b a b i l i t y P(Yla) is for t h e distribution of the spatial c o o r d i n a t e (x) (not the w a v e function 0(x)). Further, he a s s u m e s that

the probability distribution is translation-invariant

p(y i~) space coordinate) p(xlot ) translation invari~mce P(x - o0. He d o e s this so t h a t the derivative d / d ~ which o c c u r s in t h e F i s h e r information can b e r e p l a c e d b y the c o o r d i n a t e derivative - d / d x . This brings the F i s h e r information into a form m o r e similar to that o f a p h y s i c a l Lagrangian. Invariances in p h y s i c s a r e closely related to conservation laws. In t h e p r e s e n t case, translation-invariance suggests m o m e n t u m conservation, which requires V ( x ) = constant, o r dV/dx = 0 (no forces). If V(x) is n o t constant, is translation-invariance o f P(xla ) r e a s o n a b l e to a s s u m e ? I claim not. Think of the h a r m o n i c oscillator, with ZV(x) = ~1k x 2. Is it likely t h a t P(xla ) h a s the s a m e form w h e n a ~ 0 as w h e n lal is v e r y l a r g e ? . T o put the question in even m o r e s t a r k terms, c o n s i d e r the particle in a box, so t h a t V(x) = 0, 0 < x < L, b u t V(x) = "0r x -< 0 a n d L -< x. Surely P(xla = O) r

P(xla = ~' L) r P ( x l a = L). Frieden deals with the s e c o n d a n d third p r o b l e m s together. He introduces a s e c o n d type of information, a "bound Fisher information" J. This functional s o m e h o w describes the physical informarion which is intrinsic to the quantity measured. Although he strenuously tries to put great distance b e t w e e n J and constraints on physical systems which are invariably t r e a t e d with Lagrange multipliers, there is in fact no difference. He argues that the transfer of information b e t w e e n I a n d J during a m e a s u r e m e n t is the same: 81 = 8J. More specifically, 8(1 - J ) = 0. In summary, m y very strong reservations a b o u t F r i e d e n ' s t e c h n i c a l program to r e c o n s t r u c t p h y s i c s from F i s h e r information are t w o in number: the a s s u m p t i o n o f translation-invariance o f the p r o b a b i l i t y distribution function P(xlo0 is incorrect, and t h e c o n s t r a i n t t e r m s in the "bound information" J are p u t in in an a d hoc mann e r to guarantee t h a t the a p p r o p r i a t e laws are recovered. A physicist w o u l d a d m i t this up front. It is difficult to discern that the a u t h o r of the p r e s e n t w o r k is guilty of this, b u t he is.

VOLUME 22. NUMBER 4. 2000

73

Frieden is forced to identify the parameters of the probability distribution a with space-time coordinates (x,t) in order to introduce spatial (and temporal) derivatives into the expression for the Fisher information. As a result, he is forced to focus on uncertainties in position and time, rather than on the amplitudes of the electric and magnetic fields, or the complex wave functions of Quantum Mechanics. As a consequence, his interpretations of physical phenomena differ in significant ways from the standard interpretations o]Fphysicists. This divergence i n viewpoints is well illustrated in Fig. 1.4 in this book. This shows a point source located in a screen at the left of the page, and a diffraction pattern produced by this point source on a screen at the right side of the page. Every physicist has done this experiment. To us, the physics lies in the peaks and valleys (intensity maxima and minima), and in particular, in the ratios of heights of successive peaks, and the ratio of heights of adjacent peaks and valleys. We know that the pattern may be offset from its ideally predicted position because of slight displacements of the point source or placement and/or disorientation of the intermediate lens. The offset of the pattern is not important, this is an "engineering" problem. For us, the physics lies in the intensity distribution. For Frieden, the physics lies in the offset. There is more to physics than dynamical laws of motion. Frieden finds that Newton's Second Law of Motion is a consequence of variation of the Lagrangian L(x,5;) = T m.x 2 Y(x) (provided F = -VV), but where is Newton's First Law, whose purpose is to define the subset of reference frames which are inertial, in which the Second Law is true? Or Newton's Third Law, the conservation of momentum? Can the principles of unitarity, equivalence, covariance, or the conservation of energy, momentum, angular momentum be consequences of information of any kind? There is in fact an important role that information theory can play in the formulation of physical theories. This can be illustrated in terms of the elec-

74

THE MATHEMATICALINTELLIGENCER

tromagnetic field. The field can be formulcted in two ways, called for simplicity the 19th-century formulation and the 20th-century formulation. In the former the electric and magnetic fields, E(x,t) and B(x,t), are introduced. Then a system of 4 equations is introduced (by Maxwell) which these fields satisfy. This formulation has been called "manifestly covariant." The 20th-century formulation regards the electromagnetic field as composed of photons with two polarization (helicity) states. There is essentially a 1-1 correspondence between electromagnetic fields and superpositions of photon states. The superpositions satisfy no constraints. So: what role do Maxwell's equations play? Maxwell's equations are expressions of our ignorance. By introducing fields E(x,t) and B(x,t) we are introducing mathematical functions some of which cannot represent real physics. The function of Maxwell's equations is to eliminate all those mathematical functions which describe nonphysical fields, and to allow only those functions which do describe physically allowable fields. The simplest way to see this is as follows. Resolve both the manifestly covariant (19th-century) description and the quantum (photon) description in terms of their propagation direction 4-vectors (k,k4), where k.k - c2k24 = 0 in free space. For each 4-vector (k, k4) there are 6 amplitudes Ei(k, k4), Bi(k, k4) in the manifestly covariant description and just 2, one for each helicity, in the photon description. Choose any particular 4-vector (k, k4) and compare the transformation properties under the Poincar6 group of the 6 amplitudes from the first description and the 2 amplitudes from the second. This comparison identifies the 4 linear combinations of amplitudes from the first description which must vanish, and the 2 which describe the positive and negative photon helicities. Now transform this identification to any other 4-vector (k',k~) by a Poincar6 transformation. Viol& Maxwell's equations result. The point is that every time we write down an equation in physics we are expressing our ignorance. The only

purpose of an equation is to winnow out the nonphysical from the physical. Wouldn't it be more elegant to build up every allowable physical state from a small number of building blocks (e.g., photon states) which obey no constraints, so that there is a 1-1 correspondence between linear superpositions and physically allowable states? If there is a way that all/most/some parts of physics could be formulated in an information-theoretic way, it would be much more elegant to do it in this "building-up way" (Aufoauprinzip) than in the more classical "find-theequation-which-eliminates-the-nonphysical" approach (%earing-down way"). It may be possible to formulate some part of physics in an informationtheoretic setting. I do not believe the formulation by Frieden is successful. Department of Physics Drexel University Philadelphia, PA 19104 USA e-mail: [email protected]

NURBS:From Projective Geometry to Practical Use by Gerald E. F a r i n NATICK, MA; A K PETERS, 1999, 267 PP US $49.00, Hardcover, ISBN: 1-56881-084-9 REVIEWED BY LES PIEGL

on-Uniform Rational B-splines, commonly referred to as NURBS, have acquired a remarkable success in only a decade and a half. It all started in the late 1940's with Schoenberg's investigations into splines using truncated power functions. While research on splines remained active throughout the 50's and 60's, nothing really happened in the world of computational science of splines until the famous Cox-de Boor algorithm was published in 1972 independently by Maurice Cox and Carl de Boor. The Cox-de Boor algorithm allowed fast and reliable evaluation of B-splines without the truncated power functions and divided differences. Bill Gordon, then at Syracuse, had Rich Riesenfeld look at

N

Bdzier curves and surfaces to see how the new B-splines, defined by an easily computable recursive formula, can be used to defme curves and surfaces. Once Riesenfeld figured out the relationship between knots, nodes, and control points, he found a sche~me that was far superior to anything used thus far. It also contained B~zier curves and surfaces as special cases. In 1973 Riesenfeld's thesis was published-which marks the birth of B-spline curves and surfaces. Though these entities were quite nice to represent free-form shapes, common curves such as the circle were not representable by integral Bsplines. In 1975 Versprille's thesis became available, investigating B-spline curves and surfaces in homogeneous space. Upon projection of the 4-D functions to 3-D, a rational form was obtained, which is what we call today a rational B-spline. The work of Riesenfeld and Versprille became the basis of industrial research in the late 1970's. The CAD/CAM industry was looking for a mathematical form that was able to handle both freeform as well as specialized curves and surfaces. Companies such as Boeing aad SDRC (Structural Dynamics Research Corporation) played a crucial role in pushing this technology forward. The first commercial product based entirely on rational B-splines, called GEOMOD, was released by SDRC in the early 1980's. The term NURBS was coined around that time (probably by Bob Blomgren, working for Boeing at the time). Today NURBS are the de facto standards for geometry representation and data exchange, and are used almost exclusively in the broad field of computer-aided design and manufacturing (CAD/CAM). Farin's book on NURBS is a tremendous disappointment. In the 250 pages of text, he devotes exactly 14 pages to B-splines, not NURBS! He gives very brief (a page or two) discussions on such general topics as knot insertion, the de Boor algorithm, blossoms, and derivatives. There is nothing in these pages that the designer of a NURBS system can use. The chapter does not even teach the reader how to evaluate a NURBS curve or surface.

The rest of the book is a collection of (again very short) chapters on projective geometry, conics, B~zier formulation, Pythagorean curves, rectangular patches, rational B6zier triangles, quadrics, and Gregory patches. For the mathematician who wants to learn the basis of NURBS for further research, this book is a dead end. For the serious implementer who needs algorithmic details, this book is a waste of time. Though it might provide entertaining reading for someone with a solid knowledge of high school algebra and may even create the impression that the reader has learned something, it gives nothing of substance to think about. Department of Computer Science Universityof South Florida 4202 Fowler Avenue Tampa, FL 33620 e-mail: [email protected]

A Panoramaof Harmonic Analysis by Stephen Krantz CARUS MATHEMATICAL MONOGRAPH NUMBER 27 WASHINGTON, D,C.: THE MATHEMATICAL ASSOCIATION OF AMERICA, 1999, 368 PP, US $39.95, ISBN: 0883850311

REVIEWED BY MARSHALL ASH

I had some trouble with the topology section of the University of Chicago's master of science exam in May 1963. But the analysis section went very well. No small part of the credit for the latter result was due to Antoni Zygmund who had taught me real analysis I and II, and to Alberto Calder6n who had taught me complex analysis I and II. In very short order, I abandoned my plan of specializing in point-set topology and picked Fourier series as the main topic for my next hurdle, the two-topic exam. Over the next year, Bill Connant, Larry Doruoff, and I read Zygmund's Trigonometric Series in preparation for the exam. After the exam, Professor Zygmund accepted me as his student and my career as a harmonic analyst began.

Looking through Krantz's A Panorama of Harmonic Analysis feels like watching a home movie production entitled "The Zygmund School of Analysis: 1965-1999." Throughout most of this period, I was lucky enough to be near the University of Chicago, where the Monday 3:45 PM CalderduZygmund seminar featured, among many other things, just about every development in harmonic analysis mentioned in Krantz's book. These were exciting times in harmonic analysis, and my connection with the University of Chicago's seminar placed me near the center of the action. I have always been attracted by questions that have crisp, easily grasped statements. A good example of such was Lusin's conjecture that there could exist a real-valued square-integrable function (termed on the iffterval ~ = [0,2 ~r) whose Fourier series diverged at each point of a set of positive measure. Already in 1927 Kolmogorov had given an example where the function was integrable, but not square-integrable, and the Fourier series diverged at every point. Since giving an example seemed like it couldn't be very hard, I proposed to Zygmund that I take the Lusin conjecture for my thesis problem. He immediately discouraged this idea, explaining that this problem might prove to be rather difficult. Zygmund had realized that the square-integrable case was much deeper than the integrable case, even though his almost infallible intuition this time predicted the existence of an example. In the early fall of 1964 my fellow graduate student Lance Small came back from a summer visit to Berkeley carrying the news that Lennart Carleson had just proved that the Fourier series of a square-integrable function actually converges almost everywhere. He was closely questioned by Zygmund and Calder6n, who thought that he could not have gotten the story straight. But he had and Carleson had. Several years later when I spent two months working through Richard Hunt's careful exposition of his extension of Carleson's theorem, it became clear to me that although Zygmund's guess about the outcome of the conjecture had been wrong, his assessment that an extremely high level

VOLUME22, NUMBER4, 2000 7 5

o f m a t h e m a t i c s w o u l d b e required to decide the issue w a s quite accurate. After finishing m y degree in 1966, I was a Ritt instructor at Columbia. I was at a loss for h o w to begin m y c a r e e r as a r e s e a r c h mathematician. My inertia was assisted by the cultural c o r n u c o p i a that New York City provided, a n d also b y the Columbia c a m p u s p r o t e s t movem e n t featuring Mark Rudd, SDS, and the o c c u p a t i o n o f t h e m a t h building which contained m y office. I w r o t e to Zygmund, who s u g g e s t e d that I get into partial differential equations. This certainly pr()ved prophetic. The great bulk of harmonic a n a l y s i s being done n o w s e e m s to be in c o n n e c t i o n with partial differential equations. One of the w a y s to see this is to n o t e t h a t the preponderance of talks being given n o w a d a y s at the CalderSn-Zygmund s e m i n a r fits this profile. Nevertheless, m o s t of m y o w n interest never did m o v e in that direction. One thing I did to stay mathematically alive w a s to a t t e n d Stein's semin a r at Princeton. One o f the talks I h e a r d t h e r e w a s b y Stein's e x t r e m e l y young Ph.D. s t u d e n t Charles Fefferman. At the time, I d i d n o t have a sufficient overview o f h a r m o n i c analysis to a p p r e c i a t e the d e p t h a n d b e a u t y of his mathematics, b u t fortunately I have h e a r d him lecture m a n y times since. It is also fortunate t h a t his e x p o s i t o r y skills have i m p r o v e d f r o m very g o o d to extraordinary. F o r e x a m p l e , I c o n s i d e r it a high c o m p l i m e n t w h e n I s a y that Krantz's b o o k d o e s j u s t i c e to the lectures I later h e a r d in Chicago, w h e r e i n F e f f e r m a n e x p l a i n e d the p r o o f o f his t h e o r e m that the c h a r a c t e r i s t i c function o f the unit ball is n o t a multiplier on Lp(R2) w h e n p r 2. After three y e a t s at Columbia, I moved to DePaul a n d b a c k to the CalderSn-Zygmund seminar. When I first arrived in Chicago, the harmonic analysts there were reading Igari's b o o k on multiple Fourier series.[I] Grant WeUand and I immediately began doing research in this direction, and the main thrust of m y mathematical career has been in this direction ever since. F o r this reason I have an especially strong interest in chapter 3 o f A Panorama of H a r m q n i c Analysis, which is entitled

Multiple Fourier Series.

76

THE MATHEMATICAL INTELLIGENCER

Extending the w o r k o f Carleson, R i c h a r d Hunt p r o v e d that t h e F o u r i e r s e r i e s o f an L p ( y ) function converges a l m o s t everywhere, p r o v i d e d that p > 1. W h a t h a p p e n s in d i m e n s i o n two? Krantz points out that if "converges" m e a n s that the partial s u m s include t e r m s o f the series with indices lying in the dilates of a fLxed polygon, the analogue o f Hunt's T h e o r e m is true, w h e r e a s if "converges" m e a n s that the p a r t i a l sums are t a k e n to include the t e r m s with indices lying in rectangles o f variable eccentricity, t h e n t h e r e is a c o u n t e r e x a m p l e , due to Charles Fefferman. (Larry Gluck and I l a t e r a d d e d a s m a l l "bell and whistle" to that example.) But the m o s t i m p o r t a n t question o f w h a t h a p p e n s w h e n "converges" m e a n s that the partial s u m s include the t e r m s with indices lying in t h e dilates o f a n origin-centered disk r e m a i n s unsolved. Fefferman's T h e o r e m that the unit ball is not a multiplier guarantees that it is not enough for p to b e greater t h a n 1, b u t gives no insight a s to w h a t h a p p e n s when p = 2. This l e a v e s open the question of w h e t h e r the F o u r i e r series o f an L2('0-2) function h a s circularly c o n v e r g e n t partial sums a l m o s t everywhere. To m y w a y of thinking, this question is the Mount E v e r e s t o f multiple F o u r i e r series. A n interesting question n o t dealt w i t h in c h a p t e r 3 is the question o f uniqueness. Is the t r i g o n o m e t r i c series with every coefficient equal to zero the only one that converges at e v e r y p o i n t to 0? I have s p e n t m u c h o f m y life w o r k i n g on this question a n d have b e e n p l e a s e d to s e e an a l m o s t comp l e t e set of a n s w e r s discovered.[AW] The only thing I w a n t to s a y h e r e is that u n i q u e n e s s has b e e n s h o w n to hold in m a n y cases, b u t here the situation is o p p o s i t e to that for c o n v e r g e n c e of F o u r i e r series m e n t i o n e d above. We do k n o w that uniqueness h o l d s for circularly c o n v e r g e n t d o u b l e t r i g o n o m e t r i c series, b u t w e don't k n o w if it h o l d s for square convergent double trigonometric series. Speaking o f c h a p t e r 3, one thing I w o u l d like to clarify is the definition o f r e s t r i c t e d r e c t a n g u l a r convergence. I tried to explain this very s u b t l e defmition in m y 1971 p a p e r with Welland, a n d I will t a k e a n o t h e r t r y at it here.

Fix

a

large

n u m b e r E > > 1. Let {amn}m=l,2,...;n=l,2.... be a doubly i n d e x e d series o f c o m p l e x n u m b e r s and d e n o t e their r e c t a n g u l a r partial s u m s b y S M N = ~Mm= 1 ~ N n=l amn. Then s a y t h a t S = ~amn i s E-restrictedly rectangularly c o n v e r g e n t to the c o m p l e x numb e r s(E) if

lira M , N -->

S M N -~- s ( Z ) .

~1 < y N such that n 2 -- N. Thus if a n y eccentricity E is given, as s o o n as N exc e e d s E, t h e condition M/N < E bec o m e s i n c o m p a t i b l e with SMN :/: O. In o t h e r words, s(E) is 0 for every E, s o that this s e r i e s is restrictedly rectangularly c o n v e r g e n t to 0. A n d this happ e n s d e s p i t e the fact that SN2,N-1 = N, so that linlmin{M,N} ---) ~ SMNd o e s n o t exist, w h i c h is to s a y that S is not unrestrictedly r e c t a n g u l a r l y convergent. Krantz h a s m a d e w o n d e r f u l selection c h o i c e s for all o f his chapters. The c h a p t e r titles are: overview o f m e a s u r e t h e o r y a n d functional analysis, F o u r i e r series basics, the F o u r i e r transform, multiple F o u r i e r series, spherical harmonics, fractional integrals singular integrals a n d H a r d y spaces, m o d e r n theories of integral operators, wavelets, and a retrospective. In particular, I think that ending with a c h a p t e r on wavelets r e p r e s e n t s a c o r r e c t analysis of which w a y a g o o d p a r t of the w i n d s o f h a r m o n i c analysis have b e e n blowing for the p a s t few y e a r s as well as a s h r e w d g u e s s as to w h i c h w a y t h e y will b l o w in the n e a r future. A b o t a n i s t recently a s k e d m e for s o m e help in finding a g o o d m a t h e m a t i c a l r e p r e s e n t a tion for ferns that she has b e e n studying. Although m y w o r k is usually n o t v e r y applied, I have l o o k e d into this a little bit and it s e e m s likely t h a t wavelets m a y p r o v e to be the right tool.

A Panorama of Harmonic Analysis is Carus Mathematical Monograph number 27. The Publisher, the Mathematical Association of America, says that books in the series "are intended for the wide circle of thoughtful people familiar wi~h basic graduate or advanced undergraduate m a t h e m a t i c s . . , who wish to extend their knowledge without prolonged and critical study of the mathematical journals and treatises." Krantz has done an admirable job of carrying out the publisher's intentions. The right way to read this book is quickly, with-

out too much fussing over the details. While other books, such as those by Zygmund[Z] and Stein and Weiss[SW], are probably better for a graduate student who will need to achieve technical competence in the area, A Panorama of Harmonic Analysis provides an excellent way of obtaining a well-balanced overview of the entire subject.

Analysis and Nonlinear Differential Equations in Honor of Victor L. Shapiro, Contemporary Math., 208(1997), 35-71. [I] S. Igari, Lectures on Fourier Series in Several Variables, University of Wisconsin, Madison, 1968. [SW] E. M. Stein and G. Weiss, Introduction to Fourier Analysis

on

Euclidean Spaces,

Princeton Univ. Press, Princeton, 1971. [Z] A. Zygmund, Trigonometric Series, 2nd rev. ed., Cambridge Univ. Press, New York, 1959.

REFERENCES

[AVV] J. M. Ash and G. Wang, A survey of uniqueness questions in multiple trigonometric series, A Conference in Harmonic

Department of Mathematics DePaut University Chicago, IL 60614

Continued from p. 66 BIBLIOGRAPHY

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Bieberbach, L. (1930) Lehrbuch der Funktionentheorie, Springer Verlag, Berlin, (1st ed. 1921). B6cher, M. 1896 Cauchy's Theorem on complex integration, Bulletin of the American Mathematical Society (2) 2, 146-9. Briot, C.A.A. and Bouquet, J.C. (1859) Th6orie des fonctions doublement p#riodiques et, en particulier, des fonctions elliptiques, Paris. B~iot, C.A.A. and Bouquet, J.C. (1875) Th6orie des fonctions elliptiques, Paris. Denjoy, A. (1933), Sur les polygones d'approximation d'une courbe rectifiable, Comptes Rendus Acad. Sci. Paris 195, 29-32. Ahlfors, L.V. (1953) Complex Analysis, McGraw-Hill, New York. Goursat, E. (1884) D6monstration du th6or~me de Cauchy. Acta Mathematica 4, 197-200. Goursat, E. (1900) Sur la definition gen~rale des fonctions analytiques, d'apr~s Cauchy, Transactions of the American Mathematical Society, 1, 14-16. Heffter, L. (1902) Reelle Curvenintegration, GSttingen Nachrichten, 26-52. Heffter, L. (1930) 0ber den Cauchyschen Integralsatz, Mathematische Zeitschrift 32, 476-480. Hurwitz, A. and Courant, R. (1922) AIIgemeine Funktionentheorie und elliptische Funktionen. Springer Verlag, Berlin. Jordan, C. (1893) Cours d'analyse, 3 vols, Gauthier-Villars, Paris. Kamke, E. (1932) Zu dem Integralsatz von Cauchy, Mathematische Zeitschrift 33, 539-543. Knopp, K. (1930) Funktionentheorie, Springer Verlag, Leipzig and Berlin. Lichtenstein, L. (1910) Uber einige IntegrabilitAtsbedingungen zweigliedriger Differentialausdr0cke mit einer Anwendung auf den Cauchyschen Integralsatz, Sitzungsberichte der Mathematischen Gesellschaft, Berlin, 9.4, 84-100. Malmsten, C.J. (1865) Om definita integraler mellan imagin&ra gr&nser, Svenska Vetenskaps-Akademiens Handlingar, 6.3. Mittag-Leffler, G. (1922) Der Satz von Cauchy 0ber alas Integral einer

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Mittag-Leffier, G. (1873), F6rsSktillett nytt bevis fSr en sats inom de definita integralemas teori, Svenska Vetenskaps-Akademiens Handlingar. Mittag-Leffier, G. (1875) Beweis for den Cauchy'schen Satz, Nachrichten der K(Jniglichen Gesellschaft der Wissenschaften zu G6ttingen, 65-73. Moore, E.H. (1900) A simple proof of the fundamental Cauchy-Goursat Theorem, Transactions of the American Mathematical Society 1, 499-506. Osgood, W.F. (1928) Lehrbuch der Funktionentheorie, Teubner, Leipzig, 5th ed (1st ed. 1907). Perron, O. 1952 Alfred Pringsheim, Jahresbericht tier Deutschen Mathematiker Vereinigung 56, 1-6. Pringsheim, A. (1895a) Ueber den Cauchy'schen Integralsatz, Sitzungsberichte der math-phys. Classe der KOnigliche Akademie der Wissenschaften zu M(~nchen,,25, 39-72. Pringsheim, A. (1895b) Zum Cauchy'schen Integralsatz, as above, 295-304. Pringsheim, A. (1898) Zur Theorie der Doppel-lntegrale, Sitzungsberichte . . . M(~nchen 28, 59-74. Pringsheim, A. (1899) Zur Theorie der Doppel-lntegrale, Green'schen und Cauchy'schen Integralsatzes,Sitzungsberichte . . . MOnchen 29, 39-62. Pringsheim, A. (1901) Ueber den Goursat'schen Beweis des Cauchy'schen Integralsatzes, Transactions of the American Mathematical Society 2, 413-421. Pringsheim, A. (1903) Der Cauchy-Goursat'sche Integralsatz und seine 0bertragung auf reelle Kurven-lntegrale, Sitzungsberichte . . . MEJnchen 3,3, 673-682. Pringsheim, A. (1929) Kritische-historische Bemerkungen zur Funktionentheorie, Sitzungsberichte Bayerische Akademie der Wissenschaften zu M(~nchen, 281-295. Walsh, J.L. (1933), The Cauchy-Goursat Theorem for rectifiable Jordan curves, Proc. Nat. Acad. Sci, of USA 19, 540-541.

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