Preface | p. V |
Introduction and Overview | p. 1 |
How to Define f dg? | p. 2 |
Some Integral and Differential Equations | p. 7 |
Basic Assumptions | p. 9 |
Notation and Elementary Notions | p. 10 |
Definitions and Basic Properties of Extended Riemann-Stieltjes Integrals | p. 17 |
Regulated and Interval Functions | p. 17 |
Riemann-Stieltjes Integrals | p. 24 |
The Refinement Young-Stieltjes and Kolmogorov Integrals | p. 29 |
Relations between RS, RRS, and RYS Integrals | p. 46 |
The Central Young Integral | p. 51 |
The Henstock-Kurzweil Integral | p. 56 |
Ward-Perron-Stieltjes and Henstock-Kurzweil Integrals | p. 62 |
Properties of Integrals | p. 71 |
Relations between Integrals | p. 87 |
Banach-Valued Contour Integrals and Cauchy Formulas | p. 90 |
Notes | p. 98 |
-variation and p-variation; Inequalities for Integrals | p. 103 |
-variation | p. 103 |
Interval Functions, -variation, and p-variation | p. 116 |
Elementary Properties | p. 127 |
A Necessary and Sufficient Condition for Integral Duality | p. 148 |
Sufficient Conditions for Integrability | p. 168 |
Love-Young Inequalities | p. 173 |
Existence of Extended Riemann-Stieltjes Integrals | p. 191 |
Convolution and Related Integral Transforms | p. 196 |
Notes | p. 211 |
Banach Algebras | p. 215 |
Ideals and Normed Algebras | p. 216 |
The Spectral Radius | p. 219 |
Characters | p. 222 |
Holomorphic Functions of Banach Algebra Elements | p. 225 |
Complexification of Real Banach Algebras | p. 231 |
A Substitution Rule for the Kolmogorov Integral | p. 233 |
Notes | p. 234 |
Derivatives and Analyticity in Normed Spaces | p. 237 |
Polynomials and Power Series | p. 239 |
Higher Order Derivatives and Taylor Series | p. 251 |
Taylor's Formulas | p. 264 |
Tensor Products of Banach Spaces | p. 269 |
Notes | p. 270 |
Nemytskii Operators on Some Function Spaces | p. 273 |
Overview | p. 273 |
Remainders in Differentiability | p. 274 |
Higher Order Differentiability and Analyticity | p. 276 |
Autonomous Nemytskii Operators on Wp Spaces | p. 293 |
Nemytskii Operators on Wp Spaces | p. 300 |
Higher Order Differentiability and Analyticity on Wp Spaces | p. 323 |
Notes | p. 333 |
Nemytskii Operators on Lp Spaces | p. 335 |
Acting, Boundedness, and Continuity Conditions | p. 335 |
Hölder Properties | p. 347 |
Differentiability | p. 354 |
Higher Order Differentiability and Finite Taylor Series | p. 368 |
Examples where NF Is Differentiable and F Is Not | p. 378 |
Notes | p. 390 |
Two-Function Composition | p. 393 |
Overview; General Remarks | p. 393 |
Differentiability of Two-Function Composition in General | p. 394 |
Measure Space Domains | p. 397 |
Spaces with Norms Stronger than Supremum Norms | p. 401 |
Two-Function Composition on Wp Spaces | p. 404 |
Notes | p. 405 |
Product Integration | p. 407 |
Multiplicative Interval Functions and -Variation | p. 408 |
Product Integrals for Real-Valued Interval Functions | p. 415 |
Nonexistence for p > 2 | p. 424 |
Inequalities for Finite Products | p. 425 |
The Product Integral | p. 430 |
The Strict Product Integral | p. 432 |
Commutative Banach Algebras | p. 438 |
Integrals with Two Integrands | p. 442 |
Duhamel's Formula | p. 451 |
Smoothness of the Product Integral Operator | p. 457 |
Linear Integral Equations | p. 474 |
Integral Equations for Banach-Space-Valued Functions | p. 494 |
Notes | p. 499 |
Nonlinear Differential and Integral Equations | p. 505 |
Classical Picard Iteration | p. 505 |
Picard's Method in p-Variation Norm, 1 < p < 2 | p. 506 |
Existence and Uniqueness | p. 513 |
Boundedness and Convergence of Picard Iterates | p. 527 |
Continuity of the Solution Mappings | p. 536 |
Notes | p. 549 |
Fourier Series | p. 551 |
On the Order of Decrease of Fourier Coefficients | p. 551 |
Uniform Convergence | p. 554 |
Notes | p. 568 |
Stochastic Processes and -Variation | p. 571 |
Processes with Regulated Sample Functions | p. 571 |
Brownian Motion | p. 575 |
Martingales | p. 576 |
Gaussian Stochastic Processes | p. 579 |
Markov Processes | p. 600 |
Lévy Processes | p. 611 |
Empirical Processes | p. 615 |
Differentiability of Operators on Processes | p. 628 |
Integration | p. 629 |
Notes | p. 640 |
Appendix Nonatomic Measure Spaces | p. 645 |
References | p. 649 |
Subject Index | p. 661 |
Author Index | p. 665 |
Index of Notation | p. 669 |
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