Symmetric properties of real functions:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Dekker
1994
|
| Series: | Pure and applied mathematics
183 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIII, 447 S. |
| ISBN: | 0824792300 |
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MARC
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| adam_text | Contents
Preface iii
1 THE DERIVATIVE 1
1.1 Introduction 1
1.2 Even and Odd Properties 2
1.2.1 The Even and Odd Parts of a Function 2
1.2.2 Even and Odd Properties of a Function 3
1.2.3 Higher Order Symmetric Differences 4
1.3 Elementary Considerations 5
1.4 Riemann s Theorems 8
1.5 Schwarz Theorem 11
1.6 Auerbach Theorem 13
1.7 Borel Symmetric Derivative 15
1.8 Approximate Symmetric Derivative 17
1.9 Higher Order Symmetric Derivatives 18
1.10 Khintchine Theorem 20
2 CONTINUITY 23
2.1 Introduction 23
2.2 Even and Odd Continuity 24
2.2.1 Stein Zygmund Theorem 25
2.2.2 Even Version of Stein Zygmund Theorem 27
2.2.3 A Reduction Theorem 28
2.3 Charzyriski Theorem 32
2.3.1 Charzyriski s Lemma 32
2.3.2 £ (/) is Nowhere Dense 33
2.3.3 D(f) is Denumerable 35
2.3.4 D(f) is Scattered 38
2.3.5 The Even Analogue of Charzyriski s Theorem 40
2.4 Mazurkiewicz Auerbach Theorem 41
2.5 Wolibner Theorem 41
2.5.1 A Monotonicity Theorem 44
vii
viii Contents
2.6 Jurek Szpilrajn Example 45
2.7 Pesin Preiss Theorem 46
2.8 Local Symmetry 48
2.8.1 Points of Local Symmetry 50
2.9 Points of Discontinuity 51
2.9.1 Theorem of Tran 56
2.9.2 Theorems of Ponomarev and Chlebik 56
2.9.3 Points of Even/Odd Continuity 59
3 COVERING THEOREMS 63
3.1 Introduction 63
3.2 Symmetric Covering Relations 66
3.3 Basic Covering Theorems 68
3.3.1 Uniformization 69
3.3.2 Charzyriski s Covering Theorems 69
3.3.3 Fundamental Covering Theorem for Full Symmetric Covers 71
3.3.4 Other Variants 75
3.3.5 Freiling s Negligent Version 76
3.4 Khintchine Covering Theorem 77
3.5 Uher Covering Lemma 79
3.6 Approximate Symmetric Covering Relations 83
3.6.1 A Lemma for Measurable Covering Relations 84
3.6.2 Analysis of the Covering Properties 85
3.6.3 Partitioning Theorem 88
3.7 2 Interval Partitions 97
3.7.1 Rectangles and 2 Intervals 98
3.7.2 Geometrical Arguments 101
3.7.3 The Covering Theorem 107
3.7.4 Partitions into Squares 109
4 EVEN PROPERTIES Ill
4.1 Introduction Ill
4.2 Midpoint Linear Functions 112
4.2.1 Additive Functions 113
4.2.2 Hamel Theorem 115
4.2.3 Close to Midpoint Linear Functions 116
4.2.4 Locally Midpoint Linear Functions 118
4.3 Midpoint Convex Functions 119
4.3.1 Theorem of Blumberg Sierpiriski 120
4.3.2 Convex Sets, Anticonvex Sets 122
4.4 Uniform Smoothness Conditions 123
4.4.1 Modulus of Continuity 124
4.4.2 Theorem of Anderson and Pitt 127
Contents ix
4.4.3 Continuously Differentiable Functions 127
4.4.4 Theorem of M. Weiss and Zygmund 129
4.4.5 Almost Nowhere Differentiable Functions 131
4.4.6 Monotonic, Uniformly Smooth Functions 133
4.4.7 Some Examples 134
4.5 Boundedness 135
4.6 Symmetric Functions 137
4.6.1 Continuity Properties of Symmetric Functions 138
4.6.2 Baire Class 141
4.6.3 Approximately Symmetric Functions 143
4.6.4 Points of Symmetry 146
4.6.5 Points of Approximate Symmetry 147
4.6.6 Lp Symmetric Functions 148
4.6.7 Determining Sets of Symmetric Functions 148
4.6.8 Derivates of Symmetric Functions 150
4.6.9 Typical Symmetric Functions 151
4.7 Quasi Smooth Functions 152
4.7.1 Continuity Properties 152
4.7.2 A Nowhere Differentiable Quasi Smooth Function .... 153
4.7.3 Derivates of Quasi Smooth Functions 153
4.7.4 A Tauberian Theorem 156
4.8 Smooth Functions 159
4.8.1 Elementary Properties 159
4.8.2 Continuous, Smooth Functions 161
4.8.3 Makarov s Theorems 164
4.8.4 Measurable, Smooth Functions 165
4.8.5 Neugebauer s Example 166
4.8.6 Neugebauer Alternative 167
4.8.7 Approximately Smooth Functions 169
4.8.8 Ip^Smooth Functions 170
4.8.9 Discontinuities of Ip Smooth Functions 171
4.8.10 Discontinuities of Approximately Smooth Functions . . . 172
4.9 Super Smooth Functions 174
4.9.1 Differentiability a.e 175
4.9.2 Integral of Marcinkiewicz 177
4.9.3 Theorem of Denjoy 178
5 MONOTONICITY 183
5.1 Introduction 183
5.2 Some Basic Monotonicity Theorems 184
5.3 Splattered and Scattered Versions 185
5.3.1 Freiling Semi Scattered Theorem 188
5.3.2 Freiling Scattered Theorem 190
x Contents
5.3.3 Freiling Proof of Charzyriski Theorem 191
5.3.4 The Even Analogue of Charzyriski Theorem 194
5.4 Evans Larson Theorem 196
5.5 Mean Value Theorems 198
5.6 Freiling Rinne Theorem 200
5.7 Convexity Theorems 202
5.7.1 Further Convexity Theorems 205
5.8 Monotonicity Appendix 210
5.8.1 A Brief History of Symmetric Monotonicity Theorems . . 210
6 ODD PROPERTIES 213
6.1 Introduction 213
6.2 Symmetry 214
6.2.1 Exact Symmetry 214
6.2.2 Essential Symmetry 218
6.2.3 Exact Local Symmetry Everywhere 222
6.2.4 Points of Local Symmetry of a Set 225
6.2.5 Points of Local Symmetry of a Function 229
6.3 Symmetric Monotonicity 231
6.3.1 Everywhere Symmetric Increase 232
6.3.2 Points of Symmetric Increase 232
6.3.3 Porosity Properties 234
6.4 Symmetric Continuity 237
6.4.1 Theorem of Uher 238
6.4.2 Symmetrically Continuous Functions 241
6.4.3 Discontinuities of Symmetrically Continuous Functions . 242
6.4.4 Weak Symmetric Continuity 242
6.5 Chlebi k Theorem 245
6.6 Boundedness 245
6.7 Symmetric Lipschitz Conditions 247
7 THE SYMMETRIC DERIVATIVE 249
7.1 Introduction 249
7.2 Extreme Symmetric Derivatives 250
7.2.1 Relations Among the Derivates 251
7.2.2 Measurability of Derivates 255
7.2.3 Nonmeasurable Derivates 255
7.2.4 Baire Class of Symmetric Derivates 256
7.2.5 Borel Measurability of Derivates 257
7.2.6 Porosity Relations for Symmetric Derivates 257
7.2.7 Denjoy Relations for Symmetric Derivates 258
7.3 Symmetric Derivatives 262
7.3.1 Baire Class of Symmetric Derivatives 262
Contents xi
7.3.2 Symmetric Differentiability 264
7.3.3 Theorem of Belna, Evans and Humke 265
7.3.4 Example of Foran Ponomarev 266
7.3.5 Points of Non Symmetric Differentiability 266
7.3.6 Infinite Symmetric Derivatives 268
7.3.7 Steep Infinite Derivatives 269
7.3.8 Zero Symmetric Derivative 272
7.3.9 Stationary Sets for Symmetric Derivatives 273
7.3.10 Larson s Primitive 273
7.3.11 The Range of Symmetric Derivatives 276
7.4 Approximate Symmetric Derivative 278
7.4.1 Baire Class of Approximate Symmetric Derivative .... 278
7.4.2 Baire Class of Approximate Symmetric Derivates .... 280
7.4.3 Measurability of the Approximate Symmetric Derivative 283
7.4.4 Approximate Version of Khintchine Theorem 285
7.4.5 Relations for Monotone Functions 286
7.4.6 Typical Continuous Functions 287
7.5 Borel Symmetric Derivative 288
8 SYMMETRIC VARIATION 293
8.1 Introduction 293
8.2 First Order Symmetric Variation 294
8.2.1 Zero Variation 295
8.2.2 Finite Variation 297
8.2.3 Variation and Symmetric Derivates 300
8.3 Second Order Variation 302
8.3.1 Functions on Rectangles and 2 Intervals 302
8.3.2 Symmetric Increments 304
8.3.3 Variational Definitions 306
8.3.4 Basic Properties 309
8.3.5 Variation of a Continuous Function 313
8.3.6 Zero Variation 314
8.3.7 Differentiation and Variation 316
8.3.8 Finite Variation 320
8.3.9 Generalization of Schwarz Theorem 321
8.4 A Theorem of Denjoy 322
8.5 An Example of Skvorcov 322
9 SYMMETRIC INTEGRALS 323
9.1 Introduction 323
9.1.1 Integrals from Derivatives 325
9.2 The Ordinary Symmetric Integral 329
9.2.1 Preliminaries 329
xii Contents
9.2.2 A Symmetric Newton Integral 330
9.2.3 An Elementary Application to Trigonometric Series . . . 335
9.2.4 A Symmetric Totalization 336
9.2.5 A Symmetric Variational Integral 341
9.2.6 Symmetric Absolute Continuity 346
9.2.7 A Further Application to Trigonometric Series 351
9.2.8 A Symmetric Perron Integral 352
9.2.9 A Symmetric Riemann Integral 354
9.2.10 Variational Characterization of the (Rj) Integral .... 357
9.2.11 Lusin Type Characterization 361
9.3 The Approximate Symmetric Integral 362
9.3.1 Preliminaries 362
9.3.2 An Approximate Symmetric Newton Integral 363
9.3.3 An Application to Trigonometric Series 366
9.3.4 An Approximate Symmetric Variational Integral 368
9.3.5 Approximate Symmetric Absolute Continuity 374
9.3.6 An Approximate Symmetric Perron Integral 378
9.3.7 An Approximate Symmetric Riemann Integral 380
9.3.8 Applications to Trigonometric Series 383
9.4 Second Order Symmetric Integrals 388
9.4.1 A Second Order Symmetric Newton Integral 389
9.4.2 The Definitions and an Integrability Criterion 391
9.4.3 Properties of the Integral 394
9.4.4 Integration and Variation 397
9.4.5 An Application to Trigonometric Series 398
9.5 Incompatibilities 399
APPENDIX 403
A.I Scattered Sets 403
A.2 Scattered Baire theorem 406
A.3 A Density Computation 408
A.4 Density Points 409
A.5 Category Density Points 410
A.6 Hamel Bases 411
A.7 Weak Quasi Continuity 412
A.8 Weak Approximate Continuity 413
A.9 Baire Class 414
A.10 Goffman Theorem 414
A.11 Measurability 415
A.12 The Baire Property 417
PROBLEMS 421
Contents xiii
References 427
Index 441
|
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| spelling | Thomson, Brian S. Verfasser aut Symmetric properties of real functions Brian S. Thomson New York [u.a.] Dekker 1994 XIII, 447 S. txt rdacontent n rdamedia nc rdacarrier Pure and applied mathematics 183 Fonctions symétriques ram Symmetric functions Reelle Funktion (DE-588)4048918-8 gnd rswk-swf Symmetrie (DE-588)4058724-1 gnd rswk-swf Reelle Funktion (DE-588)4048918-8 s Symmetrie (DE-588)4058724-1 s DE-604 Pure and applied mathematics 183 (DE-604)BV000001885 183 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006491170&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Thomson, Brian S. Symmetric properties of real functions Pure and applied mathematics Fonctions symétriques ram Symmetric functions Reelle Funktion (DE-588)4048918-8 gnd Symmetrie (DE-588)4058724-1 gnd |
| subject_GND | (DE-588)4048918-8 (DE-588)4058724-1 |
| title | Symmetric properties of real functions |
| title_auth | Symmetric properties of real functions |
| title_exact_search | Symmetric properties of real functions |
| title_full | Symmetric properties of real functions Brian S. Thomson |
| title_fullStr | Symmetric properties of real functions Brian S. Thomson |
| title_full_unstemmed | Symmetric properties of real functions Brian S. Thomson |
| title_short | Symmetric properties of real functions |
| title_sort | symmetric properties of real functions |
| topic | Fonctions symétriques ram Symmetric functions Reelle Funktion (DE-588)4048918-8 gnd Symmetrie (DE-588)4058724-1 gnd |
| topic_facet | Fonctions symétriques Symmetric functions Reelle Funktion Symmetrie |
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