Title Page ......Page 3
Copyright Information ......Page 4
Preface ......Page 5
Introduction ......Page 6
Table of Contents ......Page 9
1.1. The Riesz-Thorin Theorem ......Page 11
1.2. Applications of the Riesz-Thorin Theorem ......Page 15
1.3. The Marcinkiewicz Theorem ......Page 16
1.4. An Application of the Marcinkiewicz Theorem ......Page 21
1.5. Two Classical Approximation Results ......Page 22
1.6. Exercises ......Page 23
1.7. Notes and Comment ......Page 29
2.1. Categories and Functors ......Page 32
2.2. Normed Vector Spaces ......Page 33
2.3. Couples of Spaces ......Page 34
2.4. Definition of Interpolation Spaces ......Page 36
2.5. The Aronszajn-Gagliardo Theorem ......Page 39
2.6. A Necessary Condition for Interpolation ......Page 41
2.7. A Duality Theorem ......Page 42
2.8. Exercises ......Page 43
2.9. Notes and Comment ......Page 46
3.1. The K-Method ......Page 48
3.2. The J-Method ......Page 52
3.3. The Equivalence Theorem ......Page 54
3.4. Simple Properties of A_{?.q} ......Page 56
3.5. The Reiteration Theorem ......Page 58
3.6. A Formula for the K-Functional ......Page 62
3.7. The Duality Theorem ......Page 63
3.8. A Compactness Theorem ......Page 65
3.9. An Extremal Property of the Real Method ......Page 67
3.10. Quasi-Normed Abelian Groups ......Page 69
3.11. The Real Interpolation Method for Quasi-Normed Abelian Groups ......Page 73
3.12. Some Other Equivalent Real Interpolation Methods ......Page 80
3.13. Exercises ......Page 85
3.14. Notes and Comment ......Page 92
4.1. Definition of the Complex Method ......Page 97
4.2. Simple Properties of A_{[?]} ......Page 101
4.3. The Equivalence Theorem ......Page 103
4.4. Multilinear Interpolation ......Page 106
4.5. The Duality Theorem ......Page 108
4.6. The Reiteration Theorem ......Page 111
4.7. On the Connection with the Real Method ......Page 112
4.8. Exercises ......Page 114
4.9. Notes and Comment ......Page 115
5.1. Interpolation of Lp-Spaces: the Complex Method ......Page 116
5.2. Interpolation of Lp-Spaces: the Real Method ......Page 118
5.3. Interpolation of Lorentz Spaces ......Page 123
5.4. Interpolation of Lp-Spaces with Change of Measure: p0 = p1 ......Page 124
5.5. Interpolation of Lp-Spaces with Change of Measure: p0 ? p1 ......Page 129
5.6. Interpolation of Lp-Spaces of Vector-Valued Sequences ......Page 131
5.7. Exercises ......Page 134
5.8. Notes and Comment ......Page 138
6.1. Fourier Multipliers ......Page 141
6.2. Definition of the Sobolev and Besov Spaces ......Page 149
6.3. The Homogeneous Sobolev and Besov Spaces ......Page 156
6.4. Interpolation of Sobolev and Besov Spaces ......Page 159
6.5. An Embedding Theorem ......Page 163
6.6. A Trace Theorem ......Page 165
6.7. Interpolation of Semi-Groups of Operators ......Page 166
6.8. Exercises ......Page 171
6.9. Notes and Comment ......Page 179
7.1. Approximation Spaces ......Page 184
7.2. Approximation of Functions ......Page 189
7.3. Approximation of Operators ......Page 191
7.4. Approximation by Difference Operators ......Page 192
7.5. Exercises ......Page 196
7.6. Notes and Comment ......Page 203
References ......Page 206
List of Symbols ......Page 215
Subject Index ......Page 216