| The Stieltjes Integral | |
| Introduction | p. 3 |
| Stieltjes integrals | p. 3 |
| Functions of bounded variation | p. 6 |
| Existence of Stieltjes integrals | p. 7 |
| Properties of Stieltjes integrals | p. 8 |
| The Stieltjes integral as a series or a Lebesgue integral | p. 10 |
| Further properties of Stieltjes integrals | p. 12 |
| Normalization | p. 13 |
| Improper Stieltjes integrals | p. 15 |
| Laws of the mean | p. 16 |
| Change of variable | p. 19 |
| Variation of the indefinite integral | p. 20 |
| Stieltjes integrals as infinite series; second method | p. 22 |
| Further conditions of integrability | p. 24 |
| Iterated integrals | p. 25 |
| The selection principle | p. 26 |
| Weak compactness | p. 33 |
| Fundamental Formulas | |
| Region of convergence | p. 35 |
| Abscissa of convergence | p. 38 |
| Absolute convergence | p. 46 |
| Uniform convergence | p. 50 |
| Analytic character of the generating function | p. 57 |
| Uniqueness of determining function | p. 59 |
| Complex inversion formula | p. 63 |
| Integrals of the determining function | p. 70 |
| Summability of divergent integrals | p. 75 |
| Inversion when the determining function belongs to L2 | p. 80 |
| Stieltjes resultant | p. 83 |
| Classical resultant | p. 91 |
| Order on vertical lines | p. 92 |
| Generating function analytic at infinity | p. 93 |
| Periodic determining function | p. 96 |
| Relation to factorial series | p. 97 |
| The Moment Problem | |
| Statement of the problem | p. 100 |
| Moment sequence | p. 101 |
| An inversion operator | p. 107 |
| Completely monotonic sequences | p. 108 |
| Function of Lp | p. 109 |
| Bounded functions | p. 111 |
| Hausdorff summability | p. 113 |
| Statement of further moment problems | p. 125 |
| The moment operator | p. 126 |
| The Hamburger moment problem | p. 129 |
| Positive definite sequences | p. 132 |
| Determinant criteria | p. 134 |
| The Stieltjes moment problem | p. 136 |
| Moments of functions of bounded variation | p. 138 |
| A sufficient condition for the solubility of the Stieltjes problem | p. 140 |
| Indeterminacy of solution | p. 142 |
| Absolutely and Completely Monotonic Functions | |
| Introduction | p. 144 |
| Elementary properties of absolutely monotonic functions | p. 144 |
| Analyticity of absolutely monotonic functions | p. 146 |
| Bernstein's second definition | p. 147 |
| Existence of one-sided derivatives | p. 149 |
| Higher differences of absolutely monotonic functions | p. 150 |
| Equivalence of Bernstein's two definitions | p. 151 |
| Bernstein polynomials | p. 152 |
| Definition of Grüss | p. 154 |
| Equivalence of Bernstein and Grüss definitions | p. 155 |
| Additional properties of absolutely monotonic functions | p. 156 |
| Bernstein's theorem | p. 160 |
| Alternative proof of Bernstein's theorem | p. 162 |
| Interpolation by completely monotonic functions | p. 163 |
| Absolutely monotonic functions with prescribed derivatives at a point | p. 165 |
| Hankel determinants whose elements are the derivatives of an absolutely monotonic function | p. 167 |
| Laguerre polynomials | p. 168 |
| A linear functional | p. 171 |
| Bernstein's theorem | p. 175 |
| Completely convex functions | p. 177 |
| Tauberian Theorems | |
| Abelian theorems for the Laplace transform | p. 180 |
| Abelian theorems for the Stieltjes transform | p. 183 |
| Tauberian theorems | p. 185 |
| Karamata's theorem | p. 189 |
| Tauberian theorems for the Stieltjes transform | p. 198 |
| Fourier transforms | p. 202 |
| Fourier transforms of functions of L | p. 204 |
| The quotient of Fourier transforms | p. 207 |
| A special Tauberian theorem | p. 209 |
| Pitt's form of Wiener's theorem | p. 210 |
| Wiener's general Tauberian theorem | p. 212 |
| Tauberian theorem for the Stieltjes integral | p. 213 |
| One-sided Tauberian condition | p. 215 |
| Application of Wiener's theorem to the Laplace transform | p. 221 |
| Another application | p. 222 |
| The prime-number theorem | p. 224 |
| Ikehara's theorem | p. 233 |
| The Bilateral Laplace Transform | |
| Introduction | p. 237 |
| Region of convergence | p. 238 |
| Integration by parts | p. 239 |
| Abscissae of convergence | p. 240 |
| Inversion formulas | p. 241 |
| Uniqueness | p. 243 |
| Summability | p. 244 |
| Determining function belonging to L2! | p. 245 |
| The Mellin transform | p. 246 |
| Stieltjes resultant | p. 248 |
| Stieltjes resultant at infinity | p. 249 |
| Stieltjes resultant completely defined | p. 250 |
| Preliminary results | p. 251 |
| The product of Fourier-Stieltjes transforms | p. 252 |
| Stieltjes resultant of indefinite integrals | p. 256 |
| Product of bilateral Laplace integrals | p. 257 |
| Resultants in a special case | p. 259 |
| Iterates of the Stieltjes kernel | p. 262 |
| Representation of functions | p. 265 |
| Kernels of positive type | p. 270 |
| Necessary and sufficient conditions for representation | p. 272 |
| Inversion and Representation Problems for the Laplace Transform | |
| Introduction | p. 276 |
| Laplace's asymptotic evaluation of an integral | p. 277 |
| Application of the Laplace method | p. 280 |
| Uniform convergence | p. 283 |
| Uniform convergence; continuation | p. 285 |
| The inversion operator for the Laplace-Lebesgue integral | p. 288 |
| The inversion operator for the Laplace-Stieltjes integral | p. 290 |
| Laplace method for a new integral | p. 296 |
| The jump operator | p. 298 |
| The variation of the determining function | p. 299 |
| A general representation theorem | p. 302 |
| Determining function of bounded variation | p. 306 |
| Modified conditions for determining functions of bounded variation | p. 308 |
| Determining function non-decreasing | p. 310 |
| The class Lp, p > 1 | p. 312 |
| Determining function the integral of a bounded function | p. 315 |
| The class L | p. 317 |
| The general Laplace-Stieltjes integral | p. 320 |
| The Stieltjes Transform | |
| Introduction | p. 325 |
| Elementary properties of the transform | p. 325 |
| Asymptotic properties of Stieltjes transforms | p. 329 |
| Relation to the Laplace transform | p. 334 |
| Uniqueness | p. 336 |
| The Stieltjes transform singular at the origin | p. 336 |
| Complex inversion formula | p. 338 |
| A singular integral | p. 341 |
| The inversion operator for the Stieltjes transform with ¿(t) an integral | p. 345 |
| The inversion operator for the Stieltjes transform in the general case | p. 347 |
| The jump operator | p. 351 |
| The variation of ¿(t) | p. 353 |
| A general representation theorem | p. 355 |
| Order conditions | p. 357 |
| General representation theorems | p. 360 |
| The function ¿(t) of bounded variation | p. 361 |
| The function ¿(t) non-decreasing and bounded | p. 363 |
| The function ¿(t) non-decreasing and unbounded | p. 365 |
| The class Lp, p > 1 | p. 368 |
| The function ¿(t) bounded | p. 372 |
| The class L | p. 374 |
| The function ¿(t) of bounded variation in every finite interval | p. 377 |
| Operational considerations | p. 381 |
| The iterated Stieltjes transform | p. 383 |
| Application to the Laplace transform | p. 384 |
| Solution of an integral equation | p. 387 |
| A related integral equation | p. 390 |
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