The Laplace Transform

CONTENTS

CHAPTER I
THE STIELTJES INTEGRAL
1.1. Introduction 3
1.2. Stieltjes integrals 3
1.3. Functions of bounded variation 6
1.4. Existence of Stieltjes integrals 7
1.5. Properties of Stieltjes integrals 8
1.6. The Stieltjes integral as a series or a Lebesgue integral 10
1.7. Further properties of Stieltjes integrals 12
1.8. Normalization 13
1.9. Improper Stieltjes integrals 15
1.10. Laws of the mean 16
1.11. Change of variable 19
1.12. Variation of the indefinite integral 20
1.13. Stieltjes integrals as infinite series; second method 22
1.14. Further conditions of integrability 24
1.15. Iterated integrals 25
1.16. The selection principle 26
1.17. Weak compactness 33

CHAPTER II
FUNDAMENTAL FORMULAS
2.1. Region of convergence 35
2.2. Abscissa of convergence 38
2.3. Absolute convergence 46
2.4. Uniform convergence 50
2.5. Analytic character of the generating function 57
2.6. Uniqueness of determining function 59
2.7. Complex inversion formula 63
2.8. Integrals of the determining function. 70
2.9. Summability of divergent integrals 75
2.10. Inversion when the determining function belongs to $L^2$ 80
2.11. Stieltjes resultant 83
2.12. Classical resultant 91
2.13. Order on vertical lines 92
2.14. Generating function analytic at infinity 93
2.15. Periodic determining function 96
2.16. Relation to factorial series 97

CHAPTER III
THE MOMENT PROBLEM SECTION PAGE
3.1. Statement of the problem 100
3.2. Moment sequence 101
3.3. An inversion operator 107
3.4. Completely monotonic sequences 108
3.5. Function of $L^p$ 109
3.6. Bounded functions 111
3.7. Hausdorff summability 113
3.8. Statement of further moment problems 125
3.9. The moment operator 126
3.10. The Halnburger moment problem 129
3.11. Positive definite sequences 132
3.12. Determinant criteria 134
3.13. The Stieltjes moment problem 136
3.14. Moments of functions of bounded variation 138
3.15. A sufficient condition for the solubility of the Stieltjes problem 140
3.16. Indeterminacy of solution 142

CHAPTER IV
ABSOLUTELY AND COMPLETELY MONOTONIC FUNCTIONS
4.1. Introduction 144
4.2. Elementary properties of absolutely monotonic functions 144
4.3. Analyticity of absolutely monotonic functions 146
4.4. Bernstein's second definition 147
4.5. Existence of one-sided depivatives 149
4.6. Higher differences of absolutely monotonic functions 150
4.7. Equivalence of Bernstein's two definitions 151
4.8. Bernstein polynomials 152
4.9. Definition of Gruss 154
4.10. Equivalence of Bernstein and Gruss definitions 155
4.11. Additional properties of absolutely monotonic functions 156
4.12. Bernstein's theorem 160
4.13. Alternative proof of Bernstein's theorem 162
4.14. Interpolation by completely monotonic functions 163
4.15. Absolutely monotonic functions ,vith prescribed derivatives at a point 165
4.16. Hankel determinants whose elements are the derivatives of an absolutely monotonic function 167
4.17. Laguerre polynomials 168
4.18. A linear functional 171
4.19. Bernstein's theorem 175
4.20. Completely convex functions 177

CHAPTER V
TAUBERIAN THEOREMS
5.1. Abelian theorems for the Laplace transform 180
5.2. Abelian theorems for the Stieltjes transform 183
5.3. Tauberian theorems 185
5.4. Karamata's theorem 189
5.5. Tauberian theorems for the Stieltjes transform 198
5.6. Fourier transforms 202
5.7. Fourier transforms of functions of $L$ 204
5.8. The quotient of Fourier transforms 207
5.9. A special Tauberian theorem 209
5.10. Pitt's form of Wiener's theorem 210
5.11. Wiener's general Tauberian theorem 212
5.12. Tauberian theorem for the Stieltjes integral 213
5.13. One-sided Tauberian condition. 215
5.14. Application of Wiener's theorem to the Laplace transform 221
5.15. Another application 222
5.16. The prime-number theorem 224
5.17. Ikehara's theorem 233

CHAPTER VI
THE BILATERAL LAPLACE TRANSFORM
6.1. Introduction 237
6.2. Region of convergence 238
6.3. Integration by parts 239
6.4. Abscissae of convergence 240
6.5. Inversion formulas 241
6.6. Uniqueness 243
6.7. Summability 244
6.8. Determining function belonging to $L^2$ 245
6.9. The Mellin transform 246
6.10. Stieltjes resultant 248
6.11. Stieltjes resultant at infinity 249
6.12. Stieltjes resultant completely defined 250
6.13. Prelimi nary results 251
6.14. The product of Fourier-Stieltjes transforms 252
6.15. Stieltjes resultant of indefinite integrals 256
6.16. Product of bilateral Laplace integrals 257
6.17. Resultants in a special case 259
6.18. Iterates of the Stieltjes kernel 262
6.19. Representation of functions 265
6.20. Kernels of positive type 270
6.21. Necessary and sufficient conditions for representation 272

CHAPTER VII
INVERSION AND REPRESENTATION PROBLEMS FOR THE LAPLACE TRANSFORM
7.1. Introduction 276
7.2. Laplace's asymptotic evaluation of an integral 277
7.3. Application of the Laplace method 280
7.4. Uniform convergence 283
7.5  Uniform convergence; continuation 285
7.6. The inversion operator for the Laplace-Lebesgue integral 288
7.7. The inversion operator for the Laplace-Stieltjes integral 290
7.8. Laplace method for a new integral 296
7.9. The jump operator 298
7.10. The variation of the determining function 299
7.11. A general representation theorem 302
7.12. Determining function of bounded variation 306
7.13. Modified conditions for determining functions of bounded variation 308
7.14. Determining function non-decreasing 310
7.15. The class $L^p$, $p>1$ 312
7.16. Determining function the integral of a bounded function 315
7.17. The class $L$ 317
7.18. The general Laplace-Stieltjes integral 320

CHAPTER VIII
THE STIELTJES TRANSFORM
8.1. Introduction 325
8.2. Elementary properties of the transform 325
8.3. Asymptotic properties of Stieltjes transforms 329
8.4. Relation to the Laplace transfornl 334
8.5. Uniqueness 336
8.6. The Stieltjes transforln singular at the origin 336
8.7. Complex inversion formula 338
8.8. A singular integral 341
8.9. The inversion operator for the Stieltjes transform with $\alpha(t)$ an integral 345
8.10. The inversion operator for the Stieltjes transform in the general case 347
8.11. The jump operator 351
8.12. The variation of $\alpha(t)$ 353
8.13. General representation theorem 355
8.14. Order conditions 357
8.15. General representation theorems 360
8.16. The function $\alpha(t)$ of bounded variation 361
8.17. The function $\alpha(t)$ non-decreasing and bounded 363
8.18. The function $\alpha(t)$ non-decreasing and unbounded 365
8.19. The class $L^p$, $p>1$ 368
8.20. The function $\varphi(y)$ bounded 372
8.21. The class $L$ 374
8.22. The function $\alpha(t)$ of bounded variation in every finite interval 377
8.23. Operational considerations 351
8.24. The iterated Stieltjes transform 383
8.25. Application to the Laplace transform 3S4
8.26. Solution of an integral equation 387
8.27. A related integral equation 390