Distributions and Fourier Transforms (eBook)

Front Cover 1
Distributors and Fourier Transforms 4
Copyright Page 5
Contents 8
Preface 6
Part I: Introduction 10
Chapter 1. Equicontinuous Families 12
Chapter 2. Infinite Products 16
Chapter 3. Convex Functions 19
Chapter 4. The Gamma Function 29
Chapter 5. Measure and Integration 32
Chapter 6. Hausdorff Measures and Dimension 39
Chapter 7. Product Measures 44
Chapter 8. The Newtonian Potential 47
Chapter 9. Harmonic Functions and the Poisson Integral 51
Chapter 10. Smooth Functions 58
Chapter 11. Taylor’s Formula 62
Chapter 12. The Orthogonal Group 67
Chapter 13. Second-Order Differential Operators 72
Chapter 14. Convex Sets 75
Chapter 15. Convex Functions of Several Variables 78
Chapter 16. Analytic Functions of Several Variables 86
Chapter 17. Linear Topological Spaces 90
Part II: Distributions 98
Chapter 18. Distributions 100
Chapter 19. Differentiation of Distributions 103
Chapter 20. Topology of Distributions 106
Chapter 21. The Support of a Distribution 110
Chapter 22. Distributions in One Dimension 114
Chapter 23. Homogeneous Distributions 117
Chapter 24. The Analytic Continuation of Distributions 123
Chapter 25. The Convolution of a Distribution with a Test Function 127
Chapter 26. The Convolution of Distributions 132
Chapter 27. Harmonic and Subharmonic Distributions 136
Chapter 28. Temperate Distributions 143
Chapter 29. Fourier Transforms of Functions in S 147
Chapter 30. Fourier Transforms of Temperate Distributions 153
Chapter 31. The Convolution of Temperate Distributions 158
Chapter 32. Fourier Transforms of Homogeneous Distributions 163
Chapter 33. Periodic Distributions in One Variable 170
Chapter 34. Periodic Distributions in Several Variables 174
Chapter 35. Spherical Harmonics 176
Chapter 36. Singular Integrals 184
Part III: Harmonic Analysis 188
Chapter 37. Functions of Positive Type 190
Chapter 38. Groups of Unitary Transformations 198
Chapter 39. Autocorrelation Functions 201
Chapter 40. Uniform Distribution Modulo 1 206
Chapter 41. Schoenberg’s Theorem 210
Chapter 42. Distributions of Positive Type 216
Chapter 43. Paley-Wiener Theorems 219
Chapter 44. Functions of the Pick Class 224
Chapter 45. Titchmarsh Convolution Theorem 233
Chapter 46. The Spectrum of a Distribution 236
Chapter 47. Tauberian Theorems 245
Chapter 48. Prime Number Theorem 250
Chapter 49. The Riemann Zeta Function 255
Chapter 50. Beurling’s Theorem 261
Chapter 51. Riesz Convexity Theorem 266
Chapter 52. The Salem Example 272
Chapter 53. Convolution Operators 280
Chapter 54. A Hardy-Littlewood Inequality 287
Chapter 55. Functions of Exponential Type 294
Chapter 56. The Bessel Kernel 301
Chapter 57. The Bessel Potential 306
Chapter 58. The Spaces of the Bessel Potential 315
Index 322