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Chapter 1．Some Classical Theorems 1

1.1．The Riesz-Thorin Theorem 1

1.2．Applications of the Riesz-Thorin Theorem 5

1.3．The Marcinkiewicz Theorem 6

1.4．An Application of the Marcinkiewicz Theorem 11

1.5．Two Classical Approximation Results 12

1.6．Exercises 13

1.7．Notes and Comment 19

Chapter 2．General Properties of Interpolation Spaces 22

2.1．Categories and Functors 22

2.2．Normed Vector Spaces 23

2.3．Couples of Spaces 24

2.4．Definition of Interpolation Spaces 26

2.5．The Aronszajn-Gagliardo Theorem 29

2.6．A Necessary Condition for Interpolation 31

2.7．A Duality Theorem 32

2.8．Exercises 33

2.9．Notes and Comment 36

Chapter 3．The Real Interpolation Method 38

3.1．The K-Method 38

3.2．The J-Method 42

3.3．The Equivalence Theorem 44

3.4．Simple Properties of？ 46

3.5．The Reiteration Theorem 48

3.6．A Formula for the K-Functional 52

3.7．The Duality Theorem 53

3.8．A Compactness Theorem 55

3.9．An Extremal Property of the Real Method 57

3.10．Quasi-Normed Abelian Groups 59

3.11．The Real Interpolation Method for Quasi-Normed Abelian Groups 63

3.12．Some Other Equivalent Real Interpolation Methods 70

3.13．Exercises 75

3.14．Notes and Comment 82

Chapter 4．The Complex Interpolation Method 87

4.1．Definition of the Complex Method 87

4.2．Simple Properties of ？ 91

4.3．The Equivalence Theorem 93

4.4．Multilinear Interpolation 96

4.5．The Duality Theorem 98

4.6．The Reiteration Theorem 101

4.7．On the Connection with the Real Method 102

4.8．Exercises 104

4.9．Notes and Comment 105

Chapter 5．Interpolation of Lp-Spaces 106

5.1．Interpolation of Lp-Spaces：the Complex Method 106

5.2．Interpolation of Lp-Spaces：the Real Method 108

5.3．Interpolation of Lorentz Spaces 113

5.4．Interpolation of Lp-Spaces with Change of Measure：p0=p1 114

5.5．Interpolation of Lp-Spaces with Change of Measure：p0≠p1 119

5.6．Interpolation of Lp-Spaces of Vector-Valued Sequences 121

5.7．Exercises 124

5.8．Notes and Comment 128

Chapter 6．Interpolation of Sobolev and Besov Spaces 131

6.1．Fourier Multipliers 131

6.2．Definition of the Sobolev and Besov Spaces 139

6.3．The Homogeneous Sobolev and Besov Spaces 146

6.4．Interpolation of Sobolev and Besov Spaces 149

6.5．An Embedding Theorem 153

6.6．A Trace Theorem 155

6.7．Interpolation of Semi-Groups of Operators 156

6.8．Exercises 161

6.9．Notes and Comment 169

Chapter 7．Applications to Approximation Theory 174

7.1．Approximation Spaces 174

7.2．Approximation of Functions 179

7.3．Approximation of Operators 181

7.4．Approximation by Difference Operators 182

7.5．Exercises 186

7.6．Notes and Comment 193

References 196

List of Symbols 205

Subject Index 206