Analysis IV: Integration and Spectral Theory, Harmonic Analysis, the Garden of Modular Delights

534 Pages·2015·3.03 MB·English

by  Roger Godement

Universitext
Roger Godement
Analysis IV
Integration and Spectral Theory,
Harmonic Analysis, the Garden of
Modular Delights
Universitext
Universitext
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Roger Godement
Analysis IV
Integration and Spectral Theory,
Harmonic Analysis,
the Garden of Modular Delights
Translated by Urmie Ray
Roger Godement
Paris, France
Translator: Urmie Ray

Translation from the French language edition:
Analyse mathematique IV by Roger Godement
Copyright © Springer-Verlag GmbH Berlin Heidelberg 2003
Springer-Verlag GmbH Berlin Heidelberg is part of Springer Science+Business Media
All Rights Reserved.

ISSN 0172-5939 ISSN219 1-6675 (ele ctronic)
Universitext
ISBN 978-3-319-16906-4 ISBN978 -3-319-16907-1 (eB ook)
DOI 10.1007/978-3-319-16907-1

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Table of Contents of Volume IV
XI { Integration and Fourier Transform :::::::::::::::::::::: 1
x1. The Upper Integral of a Positive Function .................... 5
1 { Integral of an lsc Function................................ 5
(i) Positive measures..................................... 5
(ii) Dini’s Theorem ...................................... 6
(iii) Integral of an lsc function ............................ 8
2 { Upper Integral of a Positive Function. Null Sets, Reasonable
Sets ................................................... 10
(i) Upper integrals....................................... 10
(ii) Null sets............................................ 13
(iii) Reasonable sets and functions......................... 14
3 { Fp Spaces.............................................. 15
(i) De(cid:12)nition of Fp spaces................................ 15
(ii) Convergence in mean and almost everywhere ............ 17
x2. Lp Spaces ................................................ 20
4 { Integrable Functions, Lp Spaces........................... 20
(i) Integral of an integrable function ....................... 20
(ii) Lpspaces; the Riesz-Fischer theorem ................... 23
(iii) The case of lsc or usc functions ....................... 26
5 { Lebesgue’s Theorems .................................... 27
(i) The dominated convergence theorem .................... 27
(ii) Relation between Lpand L1; H(cid:127)older’s inequality ......... 30
(iii) Applications to Fourier transforms on R................ 34
x3. Measurable Sets and Functions .............................. 37
6 { Measurable and Integrable Sets ........................... 37
(i) Properties of integrable sets............................ 37
(ii) Measurable sets...................................... 39
7 { Measurable Functions.................................... 41
(i) Separable spaces...................................... 41
(ii) Measurable maps .................................... 42
8 { Measurability and Continuity............................. 45
(i) Egorov’s and Lusin’s theorems ......................... 45
(ii) Lusin-measurable functions............................ 49
V
VI Table of Contents of Volume IV
9 { Measurability and Integrability ........................... 50
x4. Lebesgue-Fubini’s Way ..................................... 53
10 { The Lebesgue-Fubini Theorem (LF) ...................... 53
(i) Product of measures .................................. 53
(ii) The Lebesgue-Fubini theorem ......................... 54
(iii) Additions to the LF theorem ......................... 57
(iv) The Fourier inversion formula ......................... 61
11 {{ A Topological Interlude: Polish Spaces .................... 61
(i) Polish spaces......................................... 61
(ii) Lsc functions on a locally compact Polish space.......... 67
(iii) Borel sets in a Polish space ........................... 68
12 { Continuous Sums of Measures: Examples.................. 70
(i) Product measures .................................... 70
(ii) Measures induced by locally integrable densities ......... 70
(iii) Image of a measure under a map ...................... 71
(iv) Quotient of an invariant measure ...................... 72
13 { Integrable Functions with respect to a Continuous Sum ..... 74
(i) The case of lsc functions............................... 74
(ii) The generalized Lebesgue-Fubini theorem ............... 76
14 { Integrable Functions with respect to the Image of a Measure. 79
15 { Invariant Measures under Group Actions.................. 84
(i) Invariant measures on a group ......................... 84
(ii) Continuous linear representations ...................... 86
(iii) Quotient of a space by a group........................ 89
(iv) Quotient of an invariant measure ...................... 91
(v) An example: the orthogonal group on Rn................ 94
(vi) The case of homogeneous spaces....................... 96
(vii) The case of discrete groups........................... 98
x5. The Lebesgue-Nikodym Theorem............................. 103
16 { Measures With Respect To a Base Measure (cid:21): Integrable
Functions............................................... 103
17 { The Lebesgue-Nikodym Theorem (LN).................... 107
(i) Characterization of absolutely continuous measures ....... 107
(ii) Application to complex measures ...................... 108
(iii) Lebesgue decomposition.............................. 112
18 { Continuous Linear Functionals on Lp. The L1 Space ....... 114
x6. Spectral Decomposition on a Hilbert Space .................... 118
19 { Operators on a Hilbert Space ............................ 118
(i) De(cid:12)nitions, continuous linear functionals ................ 118
(ii) Orthonormal bases ................................... 121
(iii) Adjoints, Hermitian operators......................... 122
(iv) Spectrum of a Hermitian operator ..................... 124
(v) Weak topology....................................... 126
(vi) Hilbert-Schmidt operators ............................ 128
(vii) Von Neumann algebras .............................. 130
Table of Contents of Volume IV VII
20 { Gelfand’s Theorems on Normed Algebras.................. 134
21 { A Characterization of Algebras of Continuous Functions .... 141
22 { Spectral Decompositions ................................ 143
(i) The GN algebra of a normal operator ................... 143
(ii) Spectral measure of an operator algebra ................ 145
(iii) Integration with respect to a spectral measure .......... 146
(iv) Spectral decomposition of a normal operator ............ 151
23 { Self-Adjoint Operators.................................. 153
(i) Inverse of an injective Hermitian operator ............... 153
(ii) Canonical extension of a positive symmetric operator..... 156
24 { Continuous Sum Decomposition.......................... 162
(i) Virtual eigenvectors................................... 162
(ii) Continuous sums of Hilbert spaces ..................... 167
(iii) The L2 space of the integral of a measure .............. 170
x7. The Commutative Fourier Transform ........................ 173
25 { Convolution Product on a Lcg ........................... 173
(i) Convolutions and representations ....................... 173
(ii) Convolution of two measures .......................... 175
(iii) Convolution of a measure and a function ............... 178
(iv) Convolution of two functions.......................... 181
(v) Dirac sequences ...................................... 183
26 { Fourier Transform on L1(G) ............................. 185
(i) Characters of a commutative lcg........................ 185
(ii) The topology on the dual group ....................... 188
(iii) The canonical homomorphism G(cid:0)!Gbb ................ 191
27 { Fourier Transform on L2(G) ............................. 192
(i) The algebra A(G) and its characters .................... 192
(ii) Spectral decomposition of the regular representation...... 195
(iii) The invariant measure on the dual..................... 196
(iv) Fourier inversion formula and biduality................. 201
x8. Unitary Representations of Locally Compact Groups............ 204
28 { Further Representation Theory .......................... 204
29 { Fourier Transform on a Compact Group .................. 207
(i) Irreducible representations of central groups ............. 207
(ii) Central functions on a compact group .................. 209
(iii) Spectral decomposition of Z(G) ....................... 214
(iv) Characters of Z(G) and irreducible representations....... 217
(v) Easy generalizations ................................. 220
30 { Measures and Functions of Positive Type.................. 224
(i) Measures of positive type.............................. 224
(ii) Case of a commutative group.......................... 226
(iii) Functions of positive type ............................ 227
31 { Quasi-Regular Representations of a Unimodular Group ..... 233
(i) Central measures of positive type....................... 233
(ii) The commutation theorem ............................ 235
VIII Table of Contents of Volume IV
(iii) Traces on a Hilbert algebra ........................... 241
(iv) Case of a commutative group ......................... 246
(v) Characters of a locally compact group .................. 247
(vi) Characters of type (I) ................................ 249
32 { Discrete Components of the Regular Representation ........ 251
XII {The Garden of Modular Delights or The Opium of Math-
ematicians :::::::::::::::::::::::::::::::::::::::::::::::::: 261
x1. In(cid:12)nite Series and Products in Number Theory................ 261
1 { The Mellin Transform of a Fourier Transform............... 261
2 { The Functional Equation of the (cid:16) Function ................. 266
3 { Weil’s method for the Function (cid:17)(z) ....................... 273
x2. The series P1=cos(cid:25)nz and Pexp(cid:0)(cid:25)in2z(cid:1)................... 281
P
4 { The series 1=cos(cid:25)nz .................................. 281
5 { The Identity P1=cos(cid:25)nz =(cid:18)(z)2......................... 283
(i) The fundamental domain of (cid:0)((cid:18)) ....................... 283
(ii) A general method.................................... 285
(iii) The identity f(z)=(cid:18)(z)2 =1 .......................... 286
6 { The In(cid:12)nite Product of the Function (cid:18)(u;z) ................ 288
7 { The Reciprocity Law for Gauss Sums ...................... 291
(i) Cauchy’s method ..................................... 291
(ii) The Dirichlet method ................................ 294
(iii) The quadratic reciprocity law ......................... 295
x3. The Dirichlet Series L(s;(cid:31))................................. 298
8 { The Functional Equation of (cid:17)(z): bis ...................... 298
9 { Arithmetic Interlude..................................... 300
(i) Quotient rings ....................................... 300
(ii) The groups G(m); characters modm ................... 302
(iii) Orthogonality relations............................... 304
(iv) Gauss sums ......................................... 306
(v) Case of the unit character ............................. 308
10 { The Series (cid:18) (x;(cid:31)) and L(s;(cid:31))........................... 311
f
(i) Functional equation of (cid:18) (x;(cid:31)) ......................... 311
f
(ii) The series L(s;(cid:31)) .................................... 312
x4. Elliptic Functions.......................................... 316
11 { Liouville’s Theorems.................................... 316
12 { Elliptic Functions and Theta Series....................... 318
(i) Abel’s theorem....................................... 318
(ii) General theta functions............................... 321
(iii) Metamorphoses of the Jacobi series .................... 323
13 { Eisenstein and Weierstrass’s Point of View ................ 328
(i) Convergence of Eisenstein series ........................ 328
(ii) The Weierstrass }-function............................ 329
(iii) The series P(cid:25)2=sin2(cid:25)(u+nz) and G (z).............. 333
2
Table of Contents of Volume IV IX
(iv) Relation between } and (cid:18) functions ................... 335
1
(v) Elliptic functions with given simple poles................ 338
(vi) The functions (cid:16) and (cid:27) ............................. 340
L L
14 {Elliptic Integrals........................................ 341
(i) The (cid:12)eld of elliptic functions........................... 341
(ii) The Riemann surface of the (cid:12)eld of elliptic functions ..... 342
(iii) Addition formula.................................... 345
x5. SL (R) as a Locally Compact Group ........................ 349
2
15 { Subgroups, Invariant Measure ........................... 349
(i) Actions of SL (R) on the half-plane .................... 349
2
(ii) Automorphic forms as functions on G .................. 349
(iii) Subgroups of SL ................................... 353
2
(iv) Fixed points and eigenvalues .......................... 355
(v) Invariant measure .................................... 356
(vi) The point of view of the unit disc...................... 357
16 { The Discrete Series of Representations of SL (R) .......... 359
2
(i) Integrable holomorphic functions on the half-plane........ 360
(ii) The spaces Hp of the unit disc......................... 364
r
(iii) A theorem of Paley-Wienertype for H2(P)............. 366
r
(iv) The kernel function of H2(P) ......................... 368
r
(v) The holomorphic discrete series of irreducible representa-
tions of SL (R)...................................... 370
2
(vi) Solutions of the equation f (cid:3)! =f.................... 372
r
x6. Modular Functions: The Classical Theory..................... 376
17 { Fundamental Domain, Modular Forms .................... 376
(i) Generators of the modular group ....................... 376
(ii) Fundamental domain ................................. 376
(iii) The classical de(cid:12)nition of modular forms ............... 378
(iv) Eisenstein and Poincar(cid:19)e series ......................... 380
18 { Analogues of Liouville’s Two Theorems ................... 385
(i) The Riemann surface of SL (Z) ........................ 385
2
(ii) Zeros and poles...................................... 388
(iii)Constructionofmodularformsusing(cid:1)(z)andEisenstein
series............................................... 390
(iv) Application to elliptic functions ....................... 391
x7. Fuchsian Groups .......................................... 394
19 { Generalities on Automorphic Forms ...................... 394
(i) Two lemmas on discrete subgroups ..................... 394
(ii) Generalities on automorphic forms ..................... 397
(iii) The topology of horocycles. The Riemann surface of (cid:0) ... 401
(iv) Fuchsian groups ..................................... 404
20 { Parabolic Forms and Representations of G ................ 408
21 { Poincar(cid:19)e, Eisenstein and Maa(cid:25)-Selberg Series .............. 413
(i) Poincar(cid:19)e series ....................................... 413
(ii) Poincar(cid:19)e-Eisenstein series ............................. 419