9781441969491

Concrete Functional Calculus focuses primarily on differentiability of some nonlinear operators on functions or pairs of functions.Â This Â includes composition of two functions, and the product integral, taking a matrix- or operator-valued coefficient function into a solution of a system of linear differential equations with the given coefficients.Â For nonlinear integral equations with respect to possibly discontinuous functions having unbounded variation, existence and uniqueness of solutions are proved under suitable assumptions.Key features and topics:* Extensive usage of p-variation of functions* Applications to stochastic processes.This work will serve as a thorough reference on its main topics for researchers and graduate students with a background in real analysis and, for Chapter 12, in probability.

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 W_p Spaces	p. 293
Nemytskii Operators on W_p Spaces	p. 300
Higher Order Differentiability and Analyticity on W_p Spaces	p. 323
Notes	p. 333
Nemytskii Operators on L^p 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 N_F 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 W_p 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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Concrete Functional Calculus

1441969497

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