Principal functions:
Gespeichert in:
| Hauptverfasser: | , |
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Princeton, NJ [u.a.]
D. van Nostrand Company
1968
|
| Schriftenreihe: | The University Series in Higher Mathematics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVIII, 347 Seiten graph. Darst. |
Internformat
MARC
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| 245 | 1 | 0 | |a Principal functions |c Burton Rodin, Associate Professor of Mathematics, University of California, San Diego, California, Leo Sario, Professor of Mathematics, University of California, Los Angeles, California, in collaboration with Mitsuru Nakai, Associate Professor of Mathematics, Nagoya University, Nagoya, Japan |
| 264 | 1 | |a Princeton, NJ [u.a.] |b D. van Nostrand Company |c 1968 | |
| 300 | |a XVIII, 347 Seiten |b graph. Darst. | ||
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Datensatz im Suchindex
| _version_ | 1804115360082821120 |
|---|---|
| adam_text | CONTENTS
Preface vu
INTRODUCTION: What Abe Principal Functions? 1
Chapter 0
PREREQUISITE RIEMANN SURFACE THEORY
§1. Topology of Riemann surfaces 14
1. Definition of a Riemann surface 14
Conformal structure, 14. — Holomorphic functions, 15. — Holo
morphic mappings, 15. — Basis for conformal structure, 16. — Exam¬
ples, 16. — Bordered Riemann surfaces, 17. — Arcs, 18. — Orientation,
18. — The double, 19. — Open and closed surfaces, 19.
2. Compactifications 20
One point compactification, 20. — Topological representatives, 21. —
Boundary components, 21.
3. Homology 22
Chains, 22. — Boundary operator, 22. — Intersection numbers, 22. —
Canonical bases, 23.
§2. Analysis on Riemann surfaces 24
1. Harmonic functions 24
Harmonic functions, 24. — Subharmonic functions, 24. — The Dir
ichlet problem, 24. — Regular subregions and partitions, 25. — Directed
limits, 26. — Countability, 26.
2. Differential forms 27
Notational conventions, 27. — Differentials, 28. — A second order
differential, 29. — Exterior algebra, 29.
3. Integration 30
Line integrals, 30. — Area integrals, 31. — Stokes theorem, 31.—
Integration over open sets, 32. — The Dirichlet integral, 33. — Con¬
vergence in Dirichlet norm, 36. — A normal family criterion, 37.
Chapter I
THE NORMAL OPERATOR METHOD
§1. The Main Existence Theorem 38
1. A lemma on harmonic functions 39
The 7 lemma, 39.
2. The main theorem 40
Normal operators, 40. — The main theorem, 41. — Reduction of the
problem, 42. — An invertible operator, 43. — Existence of X, 44. —
The main theorem with estimates, 45.
xii
CONTENTS xiii
§2. Normal operators 45
1. Operators on compact bordered surfaces 46
The operator Lo, 46. — The operator Lh 46. — Partitions of /3, 47.
2. Operators on open surfaces 48
Limits of normal operators, 48. — Convergence of Loo, 48. — Con¬
sistent partitions, 50. — Convergence of Lia, 50. — Direct sum opera¬
tors, 51.
3. Convergence of principal functions 51
Convergence of operators, 51. — Convergence of pa, 52.
§3. The principal functions p0 and p: 53
1. Integral representations 53
Auxiliary functions, 53. — Integral representations, 55. — Conver¬
gence of auxiliary functions, 55.
2. Convergence of p0, Pi 56
Proof of L;a= Li, 56. — Principal functions with singularities, 57.
§4. Special topics 59
1. An integral equation 60
2. Estimates of q 61
The Poincar6 metric, 61. — Poincar6 diameter, 62. — Harmonic
metric, 65. — Harmonic diameter, 66. — Comparison, 67.
3. The space of normal operators. Ahlfors problem 67
The space N(A), 67.— The space AT(a,/S), 68. —The space M(a,P),
70. — Ahlfors problem, 71. — A counterexample, 71.
4. Principal functions and orthogonal projection 74
Weyl s lemma, 74. — Poincare type inequality, 75. — Existence proof
by orthogonal projection, 77.
Chapter II
PRINCIPAL FUNCTIONS
§1. Main Extremal Theorem 81
1. An extremal function 81
Principal functions, 81. —The extremal functional, 82. — Main
Extremal Theorem, 84.
2. Special cases 85
The functions po+Pi, 85. — Meromorphic and logarithmic poles,
85. — Regular principal functions, 86.
§2. CONFORMAL MAPPING 87
1. Parallel slit mappings 87
Principal meromorphic functions, 87. — Horizontal slit mapping,
88. —• Extremal property, 90.
2. Mapping by Po+Fi 90
Compact bordered surfaces, 90. — Boundary behavior of iV, 90. —
Convexity of the boundary, 91. — Extremal property of P0+Pi, 92. —
Extremal property of Pq—Pi, 92.
xiv CONTENTS
3. Circular and radial slit planes 93
Principal meromorphic functions Fo, Fu 93. — Mapping and extremal
properties, 93.
§3. Reproducing Differentials 94
1. The Hilbert space Tk 95
Harmonic differentials, 95. — Subspaces of I , 96.
2. Reproducing differentials 97
Basic properties, 97. — Construction of ft, 98.— Proof of the
theorem, 99.
3. Orthogonal projections 101
The space Ta, 101. —The spaces T^ and T^.^Th,, 101. —The
space Th0, 103. — The spaces T^ and T ^Th., 104. — The space
Th,e, 104. — The space Ta.e, 105.
§4. Interpolation problems 105
1. Generalities 106
2. Bordered surfaces 107
Reflection of differentials, 107. — Interpolation on bordered surfaces,
108.
3. Open regions 111
§5. The theorems of Riemann Roch and Abel 114
1. Riemann Roch type theorems 114
The classical case, 117. —An application to conformal mapping, 118.
2. Abel s theorem 120
§6. Extremal length 121
1. Fundamentals 122
Linear densities, 122. — Extremal length, 123. — Extremal and
conjugate extremal distance, 123. — Level curves, 125. — An applica¬
tion, 126. — A geometric inequality, 127.
2. Infinite extremal length 128
Basic properties, 128. — Examples, 129. — Level curves, 130. —
Relative homology, 130. — Integration, 131. — Principal functions, 131.
3. Extremal length of homology classes 133
Generalized homology, 133. — Compact surfaces, 133. — Open
surfaces, 135. — I o reproducers, 136.
Chapter III
CAPACITY, STABILITY, AND EXTREMAL LENGTH
§1. Generalized capacity functions 138
1. Construction of u 138
Induced partitions, 138. — Convergence of uni, 140. Dependence of
M(a°,«1,70,y) on a0, 142. — Boundary behavior of du, 143.
2. Construction of p 145
Construction of p(f,a1, y0,71), 145. — Capacity, 147.
CONTENTS xv
3. A maximum problem 148
Boundary components of maximal capacity, 149.
§2. Extremal length 150
1. Continuity lemma 150
Definition of SF, 3*, 150. — Continuity lemma, 151. — Restatement,
151. — Notation and terminology, 152. — Definition of IF , 152.—
Proof of (a) and (b ), 153. — Proof of (c), 155.
2. Extremal and conjugate extremal distance 156
Proof of A(3*)= || dW||2, 157.
3. Properties of u and p 158
Uniqueness of du, 158. — Monotone properties, 160. — Properties of
p, 160. — Uniqueness of dp, 162.
4. Capacities 164
Notation, 164.—Capacities, 165. — Extremal properties, 167.—
Green s function, 168.
§3. Exponential mappings of plane regions 169
1. Extremal slit annuli and disks 169
The mappings U, P, 169. — A reduction to annuli, 171. — Area of
slits, 171. — Characterization of R, 171.
2. Special cases 172
Circular slit annulus, 172. ¦— Minimal circular slit annulus, 173. —
Extremal property of Pa, 174. — Extremal radial slit annulus, 175. —
Minimal radial slit annulus, 178.
3. Generalizations 178
Removable boundary components, 179. — A dual problem, 179.
§4. Stability 180
1. Parallel slit mappings 180
The mappings P*, 180. — Extremal length properties, 182.
2. The mapping P0+Pi 183
A property of P°, 183. — Convexity theorem, 185. — Proof of (b ),
186.
3. Stability 187
Strong, weak and unstable components, 187. — A condition for weak¬
ness, 187. — A condition for strength, 188. — Extendability, 189.
4. Vanishing capacity 190
Nonuniqueness, 190. — Evans potential, 191.
Chapter IV
CLASSIFICATION THEORY
§1. Inclusion relations 193
1. Properties of principal functions 193
Reproducing differentials, 193. —Spans, 194. — Univalent func¬
tions, 194. — Capacities and extremal length, 195.
xvi CONTENTS
2. Classes of Riemann surfaces 196
Notation, 196.— Classes Ohy, 196.— Classes Oay for planar sur¬
faces, 197. — Capacities on planar surfaces, 198. — Classes Oay for
nonplanar surfaces, 198. — Parabolic surfaces, 199. — Summary, 200.
§2. Other Properties of the O classes 201
1. Normal operators and ideal boundary properties 201
Degeneracies of normal operators, 201. — Removable singularities,
202. — Properties of the ideal boundary, 204. — An example, 205.
2. Extremal distances 206
Plane regions, 209.
Chapter V
ANALYTIC MAPPINGS
§1. The proximity function 211
1. Use of principal functions 211
Construction of p(f ,fi), 211. — The proximity function s({,a), 213. —
Proof of symmetry, 213. — Boundedness from below, 214. — Bounds
for p(^,a), 215. — Proof of lemma, 216.
2. The conformal metric 216
Area of S, 217. — Curvature, 218.
§2. Analytic mappings 218
1. First Main Theorem 218
Notation, 218. — The fundamental functions, 219.
2. Second Main Theorem 221
Estimate of Ft(h), 223. — Evaluation of G2, 224.
3. Defects and ramifications 226
Admissible functions, 226. — Defect ramification relation, 227. —
Consequences, 228.
§3. Meromorphic functions 228
1. The classical case 228
Proximity function, 228. — Specializing R, 229. — Classical defect
ramification relations, 230. — Admissible functions, 230.
2. Resurfaces 231
Chapter VI
PRINCIPAL FORMS AND FIELDS ON RIEMANNIAN SPACES
§1. Principal functions on Riemannian spaces 232
1. Fundamentals of Riemannian spaces 233
Riemannian spaces, 233. — Differential forms, 234. — 0 forms, 235. —
Green s functions, 237. — Harmonic functions, 238.
2. The main theorem 239
Normal operators, 239. — The main theorem, 239. — Operators Lo
and Li, 241.
CONTENTS xvii
3. Functions with singularities 242
Construction on regular subregions, 242. — Extremum property of
?Vx, 242. — The span of Q, 243. — A convergence theorem, 244. — Non
compact regions, 245. — The span for R, 247.
4. Classification of Riemannian spaces 249
The class HD, 249. — Green s function, 249. — Harmonic measures,
250. — Capacity functions, 250. — The capacity, 252. — Completeness
and degeneracy, 253. — Other null classes, 254. — List of problems, 256.
5. Interpolation problem 258
6. Principal functions in physics 260
§2. Principal forms on locally flat spaces 261
1. p forms on regular regions 262
Tangential and normal parts, 262. — The point norm, 263.
2. Bounded principal forms 263
Locally flat spaces, 263. — Problem, 264. — Normal operators,
266. — A 3 lemma, 266. — The Main Existence Theorem, 267. — Proof
of Theorem 2B for parallel V, 267.
3. Border reduction 268
Generalized Dirichlet operator, 268. — Border reduction theorem, 269.
— Solution to Problem 3B, 271.
§3. Principal forms on Riemannian spaces 272
1. Classes of p forms 272
Weak derivatives, 272. — Subclasses of harmonic forms, 273. —
Green s formulas, 273.
2. Principal harmonic fields 274
Problem, 274. — Mam theorem, 275. — Specialization, 276. — Sphere
like components, 277. — Point singularities, 278. — Ahlfors method,
278.
3. Principal harmonic forms 279
Problem, 279. — Main theorem, 280. — Specialization, 280.
4. Principal semifields 281
Semifields, 281. — Tensor potentials, 282.
5. Generalization 283
/^¦. principal forms, 283. — Existence, 284. — System of operators,
284. — Special cases, 285.
Chapter VII
PRINCIPAL FUNCTIONS ON HARMONIC SPACES
§1. Harmonic spaces 287
1. Harmonic structures 288
Regularity of open sets, 288. — Definition of harmonic space, 288. —
Basic properties, 289. — Perron family, 290.
2. Dirichlet s problem 290
Regular points, 290.—Outer regular sets, 291.
xviii CONTENTS
3. Classification 291
The operator B, 291. — Parabolicity, 292.
§2. Harmonic functions with general singularities 292
1. Problem and its reduction 293
Problem, 293. — Reformulation, 293. — Reduction, 294.
2. Riesz Schauder theory 296
Dual operator T*, 296. — Eigenvalues, 296. — The eigenvalue 1,
297. — Invariant measure, 298.
3. Solution of Problem 1C 300
Result, 300.
4. Solution of Problem IB 300
Flux, 301. — Result, 302. — Solution of the original problem, 302.
§3. General principal function problem 302
1. Principal functions 302
Quasinormal operators, 302. — Associated operator, 303. — L flux,
303.
2. Generalized main existence theorem 304
Result, 304.
Appendix
SARIO POTENTIALS ON RIEMANN SURFACES
By Mitsuru Nakai
§1. Continuity principle 306
1. Joint continuity of s(f, a) 306
Definition of s(£,a), 306. — Continuity outside the diagonal set,
307. — Decomposition of s(£,a), 307.
2. Sario potentials 310
Potential theoretic principles, 310. — Local maximum principle,
311. — Continuity principle, 311.
3. Unicity principle 312
Uniqueness, 312.
§2. Maximum principle 313
1. Frostman s maximum principle 313
Global maximum, 313.
2. Fundamental theorem 317
Capacity, 317. — Capacitary measure, 318. — Subadditivity, 320.
3. Energy principle 321
Ninomiya s theorem, 321. — Unicity of capacitary measure, 322.
Bibliography 323
Author Index 337
Subject Index 339
|
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| series2 | The University Series in Higher Mathematics |
| spelling | Rodin, Burton 1933- Verfasser (DE-588)1118767942 aut Principal functions Burton Rodin, Associate Professor of Mathematics, University of California, San Diego, California, Leo Sario, Professor of Mathematics, University of California, Los Angeles, California, in collaboration with Mitsuru Nakai, Associate Professor of Mathematics, Nagoya University, Nagoya, Japan Princeton, NJ [u.a.] D. van Nostrand Company 1968 XVIII, 347 Seiten graph. Darst. txt rdacontent n rdamedia nc rdacarrier The University Series in Higher Mathematics Harmonic functions Riemann surfaces Riemannsche Fläche (DE-588)4049991-1 gnd rswk-swf Harmonische Funktion (DE-588)4159122-7 gnd rswk-swf Funktionentheorie (DE-588)4018935-1 gnd rswk-swf Harmonische Funktion (DE-588)4159122-7 s DE-604 Riemannsche Fläche (DE-588)4049991-1 s Funktionentheorie (DE-588)4018935-1 s Sario, Leo 1916-2009 Verfasser (DE-588)102602627X aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000567825&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Rodin, Burton 1933- Sario, Leo 1916-2009 Principal functions Harmonic functions Riemann surfaces Riemannsche Fläche (DE-588)4049991-1 gnd Harmonische Funktion (DE-588)4159122-7 gnd Funktionentheorie (DE-588)4018935-1 gnd |
| subject_GND | (DE-588)4049991-1 (DE-588)4159122-7 (DE-588)4018935-1 |
| title | Principal functions |
| title_auth | Principal functions |
| title_exact_search | Principal functions |
| title_full | Principal functions Burton Rodin, Associate Professor of Mathematics, University of California, San Diego, California, Leo Sario, Professor of Mathematics, University of California, Los Angeles, California, in collaboration with Mitsuru Nakai, Associate Professor of Mathematics, Nagoya University, Nagoya, Japan |
| title_fullStr | Principal functions Burton Rodin, Associate Professor of Mathematics, University of California, San Diego, California, Leo Sario, Professor of Mathematics, University of California, Los Angeles, California, in collaboration with Mitsuru Nakai, Associate Professor of Mathematics, Nagoya University, Nagoya, Japan |
| title_full_unstemmed | Principal functions Burton Rodin, Associate Professor of Mathematics, University of California, San Diego, California, Leo Sario, Professor of Mathematics, University of California, Los Angeles, California, in collaboration with Mitsuru Nakai, Associate Professor of Mathematics, Nagoya University, Nagoya, Japan |
| title_short | Principal functions |
| title_sort | principal functions |
| topic | Harmonic functions Riemann surfaces Riemannsche Fläche (DE-588)4049991-1 gnd Harmonische Funktion (DE-588)4159122-7 gnd Funktionentheorie (DE-588)4018935-1 gnd |
| topic_facet | Harmonic functions Riemann surfaces Riemannsche Fläche Harmonische Funktion Funktionentheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000567825&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT rodinburton principalfunctions AT sarioleo principalfunctions |