- Book
- xix, 402 p. : ill. ; 23 cm.
- Foreword vii Introduction xv 1 Fourier series: completion xvi Limits of continuous functions xvi 3 Length of curves xvii 4 Differentiation and integration xviii 5 The problem of measure xviii Chapter 1. Measure Theory 1 1 Preliminaries 1 The exterior measure 10 3 Measurable sets and the Lebesgue measure 16 4 Measurable functions 7 4.1 Definition and basic properties 27 4. Approximation by simple functions or step functions 30 4.3 Littlewood's three principles 33 5* The Brunn-Minkowski inequality 34 6 Exercises 37 7 Problems 46 Chapter 2: Integration Theory 49 1 The Lebesgue integral: basic properties and convergence theorems 49 2Thespace L 1 of integrable functions 68 3 Fubini's theorem 75 3.1 Statement and proof of the theorem 75 3. Applications of Fubini's theorem 80 4* A Fourier inversion formula 86 5 Exercises 89 6 Problems 95 Chapter 3: Differentiation and Integration 98 1 Differentiation of the integral 99 1.1 The Hardy-Littlewood maximal function 100 1. The Lebesgue differentiation theorem 104 Good kernels and approximations to the identity 108 3 Differentiability of functions 114 3.1 Functions of bounded variation 115 3. Absolutely continuous functions 127 3.3 Differentiability of jump functions 131 4 Rectifiable curves and the isoperimetric inequality 134 4.1* Minkowski content of a curve 136 4.2* Isoperimetric inequality 143 5 Exercises 145 6 Problems 152 Chapter 4: Hilbert Spaces: An Introduction 156 1 The Hilbert space L 2 156 Hilbert spaces 161 2.1 Orthogonality 164 2.2 Unitary mappings 168 2.3 Pre-Hilbert spaces 169 3 Fourier series and Fatou's theorem 170 3.1 Fatou's theorem 173 4 Closed subspaces and orthogonal projections 174 5 Linear transformations 180 5.1 Linear functionals and the Riesz representation theorem 181 5. Adjoints 183 5.3 Examples 185 6 Compact operators 188 7 Exercises 193 8 Problems 202 Chapter 5: Hilbert Spaces: Several Examples 207 1 The Fourier transform on L 2 207 The Hardy space of the upper half-plane 13 3 Constant coefficient partial differential equations 221 3.1 Weaksolutions 222 3. The main theorem and key estimate 224 4* The Dirichlet principle 9 4.1 Harmonic functions 234 4. The boundary value problem and Dirichlet's principle 43 5 Exercises 253 6 Problems 259 Chapter 6: Abstract Measure and Integration Theory 262 1 Abstract measure spaces 263 1.1 Exterior measures and Caratheodory's theorem 264 1. Metric exterior measures 266 1.3 The extension theorem 270 Integration on a measure space 273 3 Examples 276 3.1 Product measures and a general Fubini theorem 76 3. Integration formula for polar coordinates 279 3.3 Borel measures on R and the Lebesgue-Stieltjes integral 281 4 Absolute continuity of measures 285 4.1 Signed measures 285 4. Absolute continuity 288 5* Ergodic theorems 292 5.1 Mean ergodic theorem 294 5. Maximal ergodic theorem 296 5.3 Pointwise ergodic theorem 300 5.4 Ergodic measure-preserving transformations 302 6* Appendix: the spectral theorem 306 6.1 Statement of the theorem 306 6. Positive operators 307 6.3 Proof of the theorem 309 6.4 Spectrum 311 7 Exercises 312 8 Problems 319 Chapter 7: Hausdorff Measure and Fractals 323 1 Hausdorff measure 324 Hausdorff dimension 329 2.1 Examples 330 2. Self-similarity 341 3 Space-filling curves 349 3.1 Quartic intervals and dyadic squares 351 3. Dyadic correspondence 353 3.3 Construction of the Peano mapping 355 4* Besicovitch sets and regularity 360 4.1 The Radon transform 363 4. Regularity of sets when d 3 370 4.3 Besicovitch sets have dimension 371 4.4 Construction of a Besicovitch set 374 5 Exercises 380 6 Problems 385 Notes and References 389 Bibliography 391 Symbol Glossary 395 Index 397.
- (source: Nielsen Book Data)9780691113869 20160528
(source: Nielsen Book Data)9780691113869 20160528
- Foreword vii Introduction xv 1 Fourier series: completion xvi Limits of continuous functions xvi 3 Length of curves xvii 4 Differentiation and integration xviii 5 The problem of measure xviii Chapter 1. Measure Theory 1 1 Preliminaries 1 The exterior measure 10 3 Measurable sets and the Lebesgue measure 16 4 Measurable functions 7 4.1 Definition and basic properties 27 4. Approximation by simple functions or step functions 30 4.3 Littlewood's three principles 33 5* The Brunn-Minkowski inequality 34 6 Exercises 37 7 Problems 46 Chapter 2: Integration Theory 49 1 The Lebesgue integral: basic properties and convergence theorems 49 2Thespace L 1 of integrable functions 68 3 Fubini's theorem 75 3.1 Statement and proof of the theorem 75 3. Applications of Fubini's theorem 80 4* A Fourier inversion formula 86 5 Exercises 89 6 Problems 95 Chapter 3: Differentiation and Integration 98 1 Differentiation of the integral 99 1.1 The Hardy-Littlewood maximal function 100 1. The Lebesgue differentiation theorem 104 Good kernels and approximations to the identity 108 3 Differentiability of functions 114 3.1 Functions of bounded variation 115 3. Absolutely continuous functions 127 3.3 Differentiability of jump functions 131 4 Rectifiable curves and the isoperimetric inequality 134 4.1* Minkowski content of a curve 136 4.2* Isoperimetric inequality 143 5 Exercises 145 6 Problems 152 Chapter 4: Hilbert Spaces: An Introduction 156 1 The Hilbert space L 2 156 Hilbert spaces 161 2.1 Orthogonality 164 2.2 Unitary mappings 168 2.3 Pre-Hilbert spaces 169 3 Fourier series and Fatou's theorem 170 3.1 Fatou's theorem 173 4 Closed subspaces and orthogonal projections 174 5 Linear transformations 180 5.1 Linear functionals and the Riesz representation theorem 181 5. Adjoints 183 5.3 Examples 185 6 Compact operators 188 7 Exercises 193 8 Problems 202 Chapter 5: Hilbert Spaces: Several Examples 207 1 The Fourier transform on L 2 207 The Hardy space of the upper half-plane 13 3 Constant coefficient partial differential equations 221 3.1 Weaksolutions 222 3. The main theorem and key estimate 224 4* The Dirichlet principle 9 4.1 Harmonic functions 234 4. The boundary value problem and Dirichlet's principle 43 5 Exercises 253 6 Problems 259 Chapter 6: Abstract Measure and Integration Theory 262 1 Abstract measure spaces 263 1.1 Exterior measures and Caratheodory's theorem 264 1. Metric exterior measures 266 1.3 The extension theorem 270 Integration on a measure space 273 3 Examples 276 3.1 Product measures and a general Fubini theorem 76 3. Integration formula for polar coordinates 279 3.3 Borel measures on R and the Lebesgue-Stieltjes integral 281 4 Absolute continuity of measures 285 4.1 Signed measures 285 4. Absolute continuity 288 5* Ergodic theorems 292 5.1 Mean ergodic theorem 294 5. Maximal ergodic theorem 296 5.3 Pointwise ergodic theorem 300 5.4 Ergodic measure-preserving transformations 302 6* Appendix: the spectral theorem 306 6.1 Statement of the theorem 306 6. Positive operators 307 6.3 Proof of the theorem 309 6.4 Spectrum 311 7 Exercises 312 8 Problems 319 Chapter 7: Hausdorff Measure and Fractals 323 1 Hausdorff measure 324 Hausdorff dimension 329 2.1 Examples 330 2. Self-similarity 341 3 Space-filling curves 349 3.1 Quartic intervals and dyadic squares 351 3. Dyadic correspondence 353 3.3 Construction of the Peano mapping 355 4* Besicovitch sets and regularity 360 4.1 The Radon transform 363 4. Regularity of sets when d 3 370 4.3 Besicovitch sets have dimension 371 4.4 Construction of a Besicovitch set 374 5 Exercises 380 6 Problems 385 Notes and References 389 Bibliography 391 Symbol Glossary 395 Index 397.
- (source: Nielsen Book Data)9780691113869 20160528
(source: Nielsen Book Data)9780691113869 20160528
Science Library (Li and Ma)
Science Library (Li and Ma) | Status |
---|---|
Stacks | |
QA320 .S84 2005 | Unknown On reserve at Li and Ma Science Library 2-hour loan |
MATH-172-01
- Course
- MATH-172-01 -- Lebesgue Integration and Fourier Analysis
- Instructor(s)
- Gu, Yu
2. Fourier analysis : an introduction [2003]
- Book
- xvi, 311 p. : ill. ; 24 cm.
- Foreword vii Preface xi Chapter 1. The Genesis of Fourier Analysis 1 Chapter 2. Basic Properties of Fourier Series 29 Chapter 3. Convergence of Fourier Series 69 Chapter 4. Some Applications of Fourier Series 100 Chapter 5. The Fourier Transform on R 129 Chapter 6. The Fourier Transform on R d 175 Chapter 7. Finite Fourier Analysis 218 Chapter 8. Dirichlet's Theorem 241 Appendix: Integration 281 Notes and References 299 Bibliography 301 Symbol Glossary 305.
- (source: Nielsen Book Data)9780691113845 20160617
(source: Nielsen Book Data)9780691113845 20160617
- Foreword vii Preface xi Chapter 1. The Genesis of Fourier Analysis 1 Chapter 2. Basic Properties of Fourier Series 29 Chapter 3. Convergence of Fourier Series 69 Chapter 4. Some Applications of Fourier Series 100 Chapter 5. The Fourier Transform on R 129 Chapter 6. The Fourier Transform on R d 175 Chapter 7. Finite Fourier Analysis 218 Chapter 8. Dirichlet's Theorem 241 Appendix: Integration 281 Notes and References 299 Bibliography 301 Symbol Glossary 305.
- (source: Nielsen Book Data)9780691113845 20160617
(source: Nielsen Book Data)9780691113845 20160617
Science Library (Li and Ma)
Science Library (Li and Ma) | Status |
---|---|
Stacks | |
QA403.5 .S74 2003B | Unknown On reserve at Li and Ma Science Library 2-hour loan |
MATH-172-01
- Course
- MATH-172-01 -- Lebesgue Integration and Fourier Analysis
- Instructor(s)
- Gu, Yu