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Frames fo r Undergraduates

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STUDENT MATHEMATICAL LIBRARY Volume 40

Frames fo r Undergraduates

Deguang Ha n Keri Kornelso n David Larso n Eric Weber

©AM, % '

AMERICAN MATHEMATICA L SOCIET Y

Providence, Rhode Islan d

Editorial Boar d Gerald B . Follan d Bra d G . Osgoo d Robin Forma n (Chair ) Michae l Starbi r d

2000 Mathematics Subject Classification. P r i m a r y 47-01 , 42-01 , 15-01 , 42C15, 15A60 , 41-01 , 47A05 , 47A30 .

For addi t iona l informatio n an d upda te s o n th i s book , visi t w w w . a m s . o r g / b o o k p a g e s / s t m l - 4 0

Library o f Congres s Cataloging-in-Publicat io n D a t a

Frames fo r undergraduate s / Deguan g Ha n .. . [e t al.] . p. cm . — (Studen t mathematica l library , ISS N 1520-912 1 ; v. 40 )

Includes bibliographica l reference s an d inde x ISBN 978-0-8218-4212- 6 (alk . paper ) 1. Frame s (Vecto r analysis ) I . Han , Deguang , 1959 -

QA433.F73 200 7 515/.63-dc22 200706079 6

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10 9 8 7 6 5 4 3 2 1 1 2 1 1 1 0 0 9 0 8 0 7

Contents

Preface fo r Instructor s i x

Acknowledgements xii i

Introduction 1

Chapter 1 . Linea r Algebr a Revie w 5

§1.1. Vecto r Space s 5

§1.2. Base s fo r Vecto r Space s 8

§1.3. Linea r Operator s an d Matrice s 1 7

§1.4. Th e Ran k o f a Linea r Operato r an d a Matri x 2 3

§1.5. Determinan t an d Trac e 2 5

§1.6. Inne r Product s an d Orthonorma l Base s 2 7

§1.7. Orthogona l Direc t Su m 3 4

§1.8. Exercise s fro m th e Tex t 3 6

§1.9. Additiona l Exercise s 3 7

Chapter 2 . Finite-Dimensiona l Operato r Theor y 4 1

§2.1. Linea r Functional s an d th e Dua l Spac e 4 1

§2.2. Ries z Representatio n Theore m an d Adjoin t Operator s 4 3

§2.3. Self-adjoin t an d Unitar y Operator s 4 6

§2.4. Orthogona l Complement s an d Projection s 5 2

VI Contents

§2.5. Th e Moore-Penros e Invers e 5 9

§2.6. Eigenvalue s fo r Operator s 6 0

§2.7. Squar e Root s o f Positive Operator s 6 7

§2.8. Th e Pola r Decompositio n 6 9

§2.9. Trace s o f Operator s 7 2

§2.10. Th e Operato r Nor m 7 4

§2.11. Th e Spectra l Theore m 7 7



Chapter 3 . Introductio n t o Finit e Frame s 8 7

§3.1. M n-Frames 8 8

§3.2. Parseva l Frame s 9 2

§3.3. Genera l Frame s an d th e Canonica l Reconstructio n

Formula 9 8

§3.4. Frame s an d Matrice s 10 4

§3.5. Similarit y an d Unitar y Equivalenc e o f Frame s 10 9

§3.6. Fram e Potentia l 11 3

§3.7. Numerica l Algorithm s 11 8



Chapter 4 . Frame s i n R 2 12 3

§4.1. Diagra m Vector s 12 3

§4.2. Equivalenc e o f Frame s 12 5

§4.3. Uni t Tigh t Frame s wit h Fou r Vector s 12 9

§4.4. Uni t Tigh t Frame s wit h k Vector s 13 1

§4.5. Fram e Surgery : Removal s an d Replacement s 13 4

§4.6. Fundamenta l Inequalit y i n R 2 13 7



Chapter 5 . Th e Dilatio n Propert y o f Frames 14 1

Contents vn

§5.1.

§5.2.

§5.3.

§5.4.

§5.5.

Chapter

§6.1.

§6.2.

§6.3.

§6.4.

§6.5.

§6.6.

§6.7.

§6.8.

Chapter

§7.1.

§7.2.

§7.3.

§7.4.

Chapter

§8.1.

§8.2.

§8.3.

§8.4.

Chapter

§9.1.

§9.2.

§9.3.

§9.4.

Orthogonal Compression s o f Base s

Dilations o f Frame s

Frames an d Obliqu e Projection s

Exercises fro m th e Tex t

Additional Exercise s

6. Dua l an d Orthogona l Frame s

Reconstruction Formul a Revisite d

Dual Frame s

Orthogonality o f Frame s

Disjoint Frame s

Super-Frames an d Multiplexin g

Parseval Dua l Frame s

Exercises fro m th e Tex t

Additional Exercise s

7. Fram e Operato r Decomposition s

Continuity o f Eigenvalue s

Ellipsoidal Tigh t Frame s

Frames wit h a Specifie d Fram e Operato r

Exercises

8. Harmoni c an d Grou p Frame s

Harmonic Frame s

Frame Representation s an d Grou p Frame s

Frame Vector s fo r Unitar y System s

Exercises

9. Samplin g Theor y

An Instructiv e Exampl e

Sampling o f Polynomial s

Sampling i n Finite-Dimensiona l Space s

An Application : Imag e Reconstructio n

141

144

149

152

152

155

155

156

163

168

172

174

179

180

183

184

189

196

203

205

205

213

220

226

229

229

231

236

246

V l l l Contents

§9.5. Samplin g i n Infinite-Dimensiona l Space s 25 3

§9.6. Exercise s 26 3

Chapter 10 . Studen t Presentation s 26 5

§10.1. Eigenspac e Decompositio n 26 5

§10.2. Squar e Root s o f Positive Operator s 26 8

§10.3. Pola r Decompositio n 27 0

§10.4. Obliqu e Projection s an d Frame s 27 4

§10.5. Vandermond e Determinan t 27 7

Chapter 11 . Anecdotes : Fram e Theor y Project s b y

Undergraduates 28 1

Bibliography 28 7

Index o f Symbol s 29 1

Index 293

Preface fo r Instructor s

This book is intended t o provide an introduction t o finite frames , pre -sented a t a leve l suitabl e fo r student s i n th e late r stage s o f thei r un -dergraduate studies . Thi s boo k evolve d fro m th e result s o f Deguan g Han an d Dav e Larso n i n [42] , whic h forme d th e kerne l fo r materia l about frame s presente d i n a n NSF-sponsore d RE U (Researc h Expe -rience fo r Undergraduates ) a t Texa s A&M University entitle d Matrix Analysis and Wavelets. Th e RE U wa s led by Dave Larso n whil e Eri c Weber an d Ker i Kornelso n acte d a s mentor s an d co-instructors .

The REU students found frame s particularl y enticin g because th e questions can be approached usin g techniques from linea r algebra an d geometry. The y wer e excite d b y th e fac t tha t ther e ar e interestin g open question s regardin g frame s i n finite dimensions , an d eve n mor e enthusiastic onc e the y realize d tha t frame s hav e widesprea d applica -tions to a variety of mathematical, scientific , an d industria l problems . We wished to develop a resource which would help undergraduate stu -dents gai n acces s t o thi s burgeonin g are a o f mathematics .

The stud y o f finite frame s provide s a n excellent , highl y motivat -ing framewor k (pardo n th e pun ) i n whic h t o introduc e th e essential s of matri x analysi s an d th e rudiment s o f finite-dimensional operato r theory. Student s lear n the mathematica l theor y whil e simultaneousl y seeing how operator theor y i s used t o develop a body o f results abou t frames. Man y o f th e 5 0 o r s o student s wh o hav e participate d i n

ix

X Preface fo r Instructor s

our RE U sit e over the pas t si x years have found thi s pure/applied ap -proach to matrix analysis exciting, especially when the applied featur e led t o publishabl e ne w results .

The prerequisite s w e woul d recommen d fo r thi s boo k ar e a one -semester cours e i n colleg e leve l linea r algebra , an d a one-semeste r proof-oriented cours e i n analysis . Th e analysi s cours e i s primaril y a maturit y requirement , i n th e sens e tha t a studen t shoul d b e wel l acquainted wit h readin g an d creatin g mathematica l proofs .

We have writte n thi s boo k wit h thre e sort s o f usages i n mind .

(1) Thi s boo k ca n serv e a s th e primar y tex t fo r a n undergrad -uate special-topic s cours e o n frames . A n earl y draf t wa s used ver y successfull y a t Grinnel l Colleg e i n suc h a course . Students wh o alread y hav e a secon d cours e i n linea r alge -bra migh t ski m ove r Chapter s 1 and 2 fairly quickly , whil e students havin g less linear algebr a experience migh t nee d t o spend mor e tim e i n thes e chapters . I n eithe r case , suc h a course migh t star t i n Chapte r 3 , flipping bac k t o th e previ -ous chapters as needed for review or to learn a necessary new topic. I n thi s case , one could reasonabl y complet e Chapter s 3, 4 , 5 , 6 , and selecte d section s from Chapter s 7 , 8 , and 9 in a one-semeste r course .

(2) W e realize tha t no t ever y mathematic s curriculu m contain s a specia l topic s cours e fo r undergraduat e students . Thi s book would als o serve very wel l for a second cours e in linea r algebra, usin g frame s a s a n applicatio n t o demonstrat e th e new theory. Th e instructor coul d spend tim e carefully goin g through th e topic s i n the firs t tw o chapters befor e introduc -ing frames . Thi s cours e migh t reasonabl y cove r Chapter s 1, 2 , 3 (section s 3. 1 throug h 3.5) , 4 , 5 , an d 6 wit h a fe w selected section s a s tim e permit s fro m th e othe r chapters .

(3) Becaus e thi s boo k wa s s o heavil y motivate d b y ou r RE U experiences, w e naturall y intende d i t t o b e a resourc e fo r summer researc h student s a s they lear n abou t fram e theor y and matri x analysis . Th e metho d use d i n th e Texa s A& M REU wa s to send the tex t t o students before th e star t o f the

Preface fo r Instructor s XI

summer program, s o that the y could start readin g and work -ing ou t th e exercise s i n th e firs t thre e chapters . Onc e th e program starte d up , th e mentor s gav e lecture s ove r th e ke y ideas and topic s in the material . Student s use d the tex t a s a reference a s the y bega n workin g o n researc h problems . W e also found tha t th e mor e advance d student s i n the progra m began volunteerin g t o presen t lectures .

The chapte r entitle d Student Presentations i s intended t o giv e a tutorial o n a selectio n o f topics , i n suc h a wa y tha t goo d student s can lear n an d compos e individua l o r smal l grou p presentation s o n the topics . Fo r instance , on e o f thes e i s a detaile d expositio n o n the proo f o f th e Pola r Decompositio n Theore m i n finite dimensions . In thi s way , student s ca n ge t involve d i n th e clas s b y actuall y givin g lectures on important topics . Th e presenters gain valuable experienc e communicating a technical proof to their peers , while giving the othe r students a change o f styl e an d pac e fro m th e instructor' s lectures .

The las t chapter , Anecdotes, tell s som e storie s abou t differen t types o f student s a s they lear n abou t frames . Wit h these , w e inten d to offer snapshot s of the various settings - classrooms , REU programs , independent studie s - i n whic h w e find student s excite d b y thi s sub -ject. Th e author s welcom e your feedbac k an d suggestions . We' d als o enjoy hearin g your "anecdotes" , sinc e we were able to share ours wit h you.

Deguang Ha n Keri Kornelso n David Larso n Eric Webe r

Acknowledgements

Our decisio n t o writ e thi s boo k aros e fro m th e succes s o f a serie s o f graduate seminars , VIGRE courses , and the NSF REU Matrix Analy-sis and Wavelets a t Texa s A&M University ove r the years 1997-2006 . We wis h t o expres s ou r thank s t o th e Nationa l Scienc e Foundatio n and to the Mathematics Department a t Texas A&M University for th e resources whic h enable d thes e seminars , courses , an d RE U program s to tak e place .

The student s wh o too k par t i n thes e activitie s a t Texa s A& M University helpe d to shape our vision and understandin g abou t fram e theory. A s we lectured, mentored , an d answere d questions , th e idea s which ar e presen t i n thi s boo k wer e taking form . Student s wh o gav e presentations o r worke d o n researc h project s relate d t o thi s mate -rial helpe d u s t o adap t th e materia l t o b e usefu l fo r undergraduat e students. W e heartily expres s ou r thank s t o al l o f these students .

Preliminary version s o f thi s boo k hav e bee n use d i n RE U pro -grams, seminars , an d course s a t Texa s A& M University , th e Univer -sity o f Centra l Florida , th e Universit y o f Iowa, Iow a Stat e University , and Grinnel l College . Student s i n thes e program s hav e foun d er -rors an d oversights , worke d ou t th e exercises , discovered duplication s or inconsistencies , an d give n u s valuabl e feedbac k o n th e book . W e greatly appreciat e thei r contributions . W e particularly recogniz e th e

xin

XIV Acknowledgement s

assistance o f Nga Nguyen wit h th e materia l i n Chapte r 4 and Sectio n 10.5.

We are grateful t o Troy Henderson, who lent his superior technica l expertise t o improv e th e figures i n thi s book .

A numbe r o f postdoctora l fellows , visitin g scholars , an d regula r faculty member s helpe d th e author s t o mentor RE U an d VIGRE stu -dents a t Texa s A& M Universit y durin g on e o r mor e years . W e tak e this opportunit y t o than k Ke n Dykema , Da n Jupiter , Robi n Harte , Marc Ordower , Davi d Redett , an d Nic o Spronk .

Thanks als o to Julie n Giol , Jimmy Dillies , David Kerr , an d Tho -mas Schlumprecht , fo r participatin g i n a workin g semina r o n frame s and operato r algebra s a t Texa s A& M Universit y wit h Dav e Larso n during 2005-2007 , an d fo r thei r comment s concernin g th e near-fina l version o f thi s RE U book . Thank s t o Kare n Shuma n fo r he r carefu l reading an d editin g o f tricky passages .

Bibliography

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[17] P . Casazza , M . Fickus , J . Kovacevic , M . Leon , an d J . Tremain , A physical interpretation of finite frames. Harmoni c Analysi s an d Appli -cations in Appl. Numer. Harmon . Anal. , Birkhaser , Boston , MA , 2006 , pp. 51-76 .

[18] P . Casazza, M . Fickus, J . Tremain , an d E . Weber, The Kadis on-Singer Problem in Mathematics and Engineering, Operato r Theory , Operato r Algebras an d Applications , Contemp . Math . 414 , Amer . Math . Soc , Providence, RI , 2006 , pp . 299-356 .

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orthogonal wavelets, Mem . Amer . Math . Soc . 13 4 (1998) , no . 640 .

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Index o f Symbol s

Symbol

(x,y) a \a\

Kj] A* AT

A-1

B(H) B(HuH2) C C(R) C[a, b] det(A) diag(ai,. • • ,a d i m F E + F ELF E + F F Fn

Description

Inner product . Complex conjugate . Modulus o f a comple x number . Components o f a matrix . Adjoint o f a n operato r o r matri x A. Transpose o f a matri x A. Inverse o f a n operato r o r matri x A. Bounded operator s o n a Hilber t space . Bounded operato r fro m Hi t o W 2-Complex numbers . Continuous function s o n R . Continuous function s o n [a , b]. Determinant o f a matrix .

n) Diagona l matrix . Dimension o f a vecto r space . Direct su m o f vector spaces . Orthogonal inne r produc t spaces . Sum o f vector spaces . Either th e rea l o r comple x numbers . n-tuples fro m F .

Page

27 5 5 18 43 20 18 74 74 5 7 6 26 19 15 16 35 16 5 6

Index o f

Symbol Descriptio n Page

F(D)

Im(a) ker(T)

P„[a,6] K Re(a) a(T)

span(£') supp(T) T > 0 T > 5

T t

T*

TA

\T\

\\T\\

v* x ®y

K*} 11x11

Space o f al l function s wit h domai n D. Orthogonal direc t su m o f Hilber t spaces . Imaginary par t o f a comple x number . Kernel o f a n operato r o r matrix . Square-summable sequence s indexe d b y / . Space o f linea r operator s fro m V to W . Orthogonal complement . Space o f m x n matrices . Space o f polynomials o f degre e < n . Polynomials restricte d t o [a , b] wit h degre e Real numbers . Real par t o f a comple x number . Spectrum o f a matri x o r operator . Transition matri x betwee n bases . Linear spa n o f a subse t o f a vecto r space . Support o f a n operator . Positive operator . Ordering o f positive operators . Moore-Penrose inverse . Square roo t o f a positiv e operator . Operator correspondin g t o a matrix A. Absolute valu e o f a n operator . Norm o f a n operator . Dual o f a vecto r space . Vector wit h respec t t o basi s B. Rank-one operator . Conjugate-transpose o f vecto r x. Canonical dua l o f frame {xi}. Norm o f a vector .

< n .

7 34 34 17 28 40 52 37 7 7 5 34 61 23 9 57 48 67 59 67 22 69 74 42 21 71 90 169 29

Index

adjoint, 43 , 44 , 45 , 49 , 6 4 analysis operator , 94 , 100 , 104 ,

111, 151 , 159 , 21 7

Banach space , 18 4 bandlimited subspace , 239 , 25 2 basis, 10 , 11-17 , 62 , 88 , 145 , 15 7

Hamel, 1 1 orthonormal, see also

orthonormal basi s transition matrix , 2 3

Beurling density , 26 2 bijection, 1 7 bounded operator , 7 4

Cauchy-Schwarz, 29 , 39 , 23 9 Chebyshev algorithm , 11 9 co-isometry, 4 8 commutant, 21 9 complete space , 3 0 compression, 141 , 14 4 condition number , 108 , 11 9 conjugate gradien t algorithm , 12 0 coordinate vector , 2 1

determinant, 27 7 diagram vector , 124 , 129 , 13 1 dilation, 14 1 dimension, 15 , 24 , 52 , 9 5 direct su m - orthogonal , 17 , 34 , 35 ,

36, 5 5

direct su m - vecto r space , 16 , 39 , 62, 265 , 27 4

discrete Fourie r transform , 143 , 237, 238 , 243 , 25 1

disjoint frames , 168 , 16 9 dual frame , 100 , 156-163 , 17 5

alternate, 15 7 canonical, 157 , 161 , 162 , 169 ,

214 Parseval, 175 , 17 7

dual space , 4 2

eigenspace, 61 , 62 , 63 , 65 , 77-8 1 eigenvalue, 60 , 62-67 , 77 , 79 , 107 ,

187, 214 , 26 5 eigenvector, 61 , 62-67 , 79 , 26 5 ellipsoid, 18 9 ellipsoidal tigh t frame , 189 , 195 ,

196 embedding, 51 , 52 equivalent frames , 12 6

PRR-equivalence, 12 7 equivalent norms , 18 4 evaluation functional , 23 5

Fourier basis , 237 , 25 1 Fourier inversio n formula , 25 6 Fourier transform , 25 5 Fourier Uncertaint y Principle , 239 ,

249

293

294 Index

frame, 2-4 , 98 , 99-112 , 142 , 145 , 177, 196 , 203 , 235 , 24 4

R n , 8 8 Parseval, see also Parseva l fram e redundancy, 103 , 21 4 tight, see also tigh t fram e

frame algorithm , 11 9 frame bounds , 98 , 105 , 115 , 14 9

optimal, 98 , 10 7 frame matrix , 10 7

Parseval fram e matrix , 10 7 tight fram e matrix , 10 7

frame operator , 100 , 104 , 110 , 123 , 156, 191 , 196 , 20 3

frame potential , 113-11 8 frame representation , 214 , 217 , 218 ,

220 frame surgery , 13 5 function

continuous, 6 functional, 41 , 4 4 fundamental inequality , 118 , 137 ,

201

geometric multiplicity , 6 1 Gram-Schmidt, 33 , 14 6 Grammian operator , 108 , 16 1 group frame , 205 , 214 , 21 7 group homomorphism , 21 3

harmonic frame , 143 , 20 6 generalized, 20 8

Hilbert space , 30 , 23 6 Hilbert-Schmidt norm , 12 2

idempotent, 14 9 injection, 17 , 1 8 inner product , 27 , 29-34 , 4 9 inner produc t space , 27 , 3 0 invariant subspace , 62 , 77 , 8 1 invertible operator , 18 , 58 , 65 , 6 9 isometry, 48 , 50 , 76 , 9 4

kernel, 17 , 24 , 5 7

Lagrange polynomials , 23 4 linear combination , 8 linear dependence , 9 linear independence , 9 , 10-12 , 3 1

linear operator , 17 , 18-25 , 41-8 1 composition, 1 8 inverse, 1 8 matrix representation , 2 2

local commutant , 221 , 22 2

matrix, 18 , 19-27 , 43 , 51 , 60, 106 , 184, 246 , 26 5

characteristic polynomial , 62 , 187

column space , 2 4 determinant, 2 6 diagonal, 1 9 diagonalizable, 6 2 full rank , 2 5 inverse, 1 9 invertible, 19 , 2 4 row space , 2 4 trace, see also trac e transpose, 2 0

Moore-Penrose inverse , 59 , 23 5 multiplexing, 17 3

norm, 2 9 Hilbert Schmidt , 18 4 maximum, 18 4

normal operator , 46 , 64 , 77 , 7 9

operator, see also linea r operator , 74

absolute value , 6 9 partial isometry , 7 1 positive, see also positiv e

operator rank-one, see also rank-on e

operator self-adjoint, see also self-adjoin t

operator square root , 67 , 68 , 110 , 268 , 26 9 unitary see also unitar y operato r

48 operator norm , 74 , 79 , 101 , 18 4 orthogonal complement , 5 2 orthogonal complementar y frame ,

147, 15 3 orthogonal frames , 164 , 166 , 17 0 orthogonal set , 3 1 orthogonal subspaces , 5 3 orthogonal vectors , 3 1

Index 295

orthonormal basis , 1-3 , 31 , 33 , 50 , 51, 56 , 95 , 97 , 142 , 145 , 206 , 258

scaled, 3 1

Parallelogram Identity , 17 0 Parseval frame , 2 , 92 , 94 , 97 , 98 ,

102, 104 , 108 , 110 , 142 , 145 , 148, 164 , 167 , 21 8

Parseval sequence , 8 9 Parseval's Identity , 2 , 3 2 partial isometry , 70 , 76 , 219 , 27 1 Payley-Weiner space , 25 3 Polar Decompositio n Theorem , 72 ,

270 Polarization Lemma , 3 3 polynomial, 7 , 3 8 positive operator , 48 , 63 , 67 , 6 8 projection

oblique, 149 , 151 , 161 , 175 , 27 4 orthogonal, 53 , 55 , 56 , 71 , 74,

76, 77 , 14 1 range, 5 7 support, 5 7

Pythagorean Theorem , 3 2

range, 1 7 rank, 2 3 rank-one decomposition , 19 8 rank-one operator , 71 , 19 0 reconstruction formula , 93 , 99 , 118 ,

155 Riesz Representatio n Theorem , 42 ,

44

sampling transform , 231 , 261 , 26 2 scalar, 5 Schur Majorizatio n Theorem , 20 1 self-adjoint operator , 46 , 47 , 53 ,

63, 67 , 77 , 26 7 set o f sampling , 231 , 235 , 244 , 26 0 set o f uniqueness , 231 , 262 similar frames , 109 , 111 , 161 , 206 similar matrices , 23 , 61 , 62 span, 9 , 10 , 1 1 spanning set , 1 0 Spectral Mappin g Theorem , 6 6 Spectral Theorem , 7 7 spectrum, 6 1 , 63-67 , 77 , 7 9

subspace, 8 , 10 , 13 , 16 , 5 2 complement, 1 6 proper, 8

super-frame, 17 4 support, 5 7 surjection, 17 , 1 8 synthesis operator , 10 4

tensor notatio n fo r operators , see also rank-on e operato r

tight frame , 98 , 115 , 123-138 , 20 1 ellipsoidal, see also ellipsoida l

tight fram e unit, 12 5

trace, 26 , 73 , 9 6 transpose, 4 3 Triangle Inequality , 2 9

uniform frame , 98 , 114 , 14 3 unitarily equivalen t frames , 109 ,

111, 146 , 20 6 unitarily equivalen t matrices , 8 6 unitarily equivalen t operators , 8 6 unitary operator , 48 , 49-51 , 63 , 76 ,

146, 21 1 unitary representation , 21 3

direct sum , 21 6 equivalent, 21 3 invariant subspace , 213 , 21 7 left regular , 216 , 21 9 right regular , 21 7 subrepresentation, 213 , 21 7

unitary system , 220 , 222 , 22 4 frame vector , 220 , 223 , 22 4

Vandermonde matrix , 233 , 27 7 vector, 6 , 8 8

zero, 6 vector space , 5 , 6-17 , 27 , 42 , 23 1

dimension, see also dimensio n finite-dimensional, 1 0 function space , 6 infinite-dimensional, 1 0 normed space , 2 9 subspace, see also subspac e superspace, 8 trivial, 6

wandering vector , 220 , 222 , 22 5

Titles i n Thi s Serie s

40 Deguan g Han , Ker i Kornelson , Davi d Larson , an d Eri c Weber , Frames fo r undergraduates , 200 7

39 Ale x losevich , A vie w fro m th e top : Analysis , combinatoric s an d numbe r theory, 200 7

38 B . Fristedt , N . Jain , an d N . Krylov , Filterin g an d prediction : A primer, 200 7

37 Svet lan a Katok , p-adi c analysi s compare d wit h real , 200 7

36 Mar a D . Neusel , Invarian t theory , 200 7

35 Jor g Bewersdorff , Galoi s theor y fo r beginners : A historica l perspective ,

2006

34 Bruc e C . Berndt , Numbe r theor y i n th e spiri t o f Ramanujan , 200 6

33 Rekh a R . Thomas , Lecture s i n geometri c combinatorics , 200 6

32 Sheldo n Katz , Enumerativ e geometr y an d strin g theory , 200 6

31 Joh n McCleary , A firs t cours e i n topology : Continuit y an d dimension ,

2006

30 Serg e Tabachnikov , Geometr y an d billiards , 200 5

29 Kristophe r Tapp , Matri x group s fo r undergraduates , 200 5

28 Emmanue l Lesigne , Head s o r tails : A n introductio n t o limi t theorerh s i n probability, 200 5

27 Reinhar d Illner , C . Sea n Bohun , Samanth a McCol lum , an d The a

van Roode , Mathematica l modelling : A cas e studie s approach , 200 5

26 Rober t Hardt , Editor , Si x theme s o n variation , 200 4

25 S . V . Duzhi n an d B . D . Chebotarevsky , Transformatio n group s fo r

beginners, 200 4

24 Bruc e M . Landma n an d Aaro n Robertson , Ramse y theor y o n th e

integers, 200 4

23 S . K . Lando , Lecture s o n generatin g functions , 200 3

22 Andrea s Arvanitoyeorgos , A n introductio n t o Li e group s an d th e

geometry o f homogeneou s spaces , 200 3

21 W . J . Kaczo r an d M . T . Nowak , Problem s i n mathematica l analysi s

III: Integration , 200 3

20 Klau s Hulek , Elementar y algebrai c geometry , 200 3

19 A . She n an d N . K . Vereshchagin , Computabl e functions , 200 3

18 V . V . Yaschenko , Editor , Cryptography : A n introduction , 200 2

17 A . She n an d N . K . Vereshchagin , Basi c se t theory , 200 2

16 Wolfgan g Kuhnel , Differentia l geometry : curve s - surface s - manifolds ,

second edition , 200 6

15 Ger d Fischer , Plan e algebrai c curves , 200 1

14 V . A . Vassiliev , Introductio n t o topology , 200 1

13 Frederic k J . Almgren , Jr. , Plateau' s problem : A n invitatio n t o varifol d geometry, 200 1

TITLES I N THI S SERIE S


II: Continuit y an d differentiation , 200 1

11 Michae l Mester ton-Gibbons , A n introductio n t o game-theoreti c modelling, 200 0

® 10 Joh n Oprea , Th e mathematic s o f soa p films : Exploration s wit h Mapl e ,

2000

9 Davi d E . Blair , Inversio n theor y an d conforma l mapping , 200 0

8 Edwar d B . Burger , Explorin g th e numbe r jungle : A journey int o

diophantine analysis , 200 0

7 Jud y L . Walker , Code s an d curves , 200 0

6 Geral d Tenenbau m an d Miche l Mende s France , Th e prim e number s

and thei r distribution , 200 0

5 Alexande r Mehlmann , Th e game' s afoot ! Gam e theor y i n myt h an d

paradox, 200 0


I: Rea l numbers , sequence s an d series , 200 0

3 Roge r Knobel , A n introductio n t o th e mathematica l theor y o f waves ,

2000

2 Gregor y F . Lawle r an d Leste r N . Coyle , Lecture s o n contemporar y

probability, 199 9

1 Charle s Radin , Mile s o f tiles , 199 9

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