Norvaiša Concrete Functional Calculus

1. Auflage 2010
ISBN: 978-1-4419-6950-7
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 671 Seiten

Concrete Functional Calculus focuses primarily on differentiability of some nonlinear operators on functions or pairs of functions.  This  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.  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.

Autoren/Hrsg.

Norvaiša, R.

Inhaltsverzeichnis

1;Preface;6
2;Contents;10
3;1 Introduction and Overview;14
3.1;1.1 How to Define sba f dg?;15
3.2;1.2 Some Integral and Differential Equations;20
3.3;1.3 Basic Assumptions;22
3.4;1.4 Notation and Elementary Notions;23
4;2 Definitions and Basic Properties of Extended Riemann– Stieltjes Integrals;29
4.1;2.1 Regulated and Interval Functions;29
4.2;2.2 Riemann–Stieltjes Integrals;36
4.3;2.3 The Refinement Young–Stieltjes and Kolmogorov Integrals;41
4.4;2.4 Relations between RS, RRS, and RYS Integrals;58
4.5;2.5 The Central Young Integral;63
4.6;2.6 The Henstock–Kurzweil Integral;68
4.7;*2.7 Ward–Perron–Stieltjes and Henstock–Kurzweil Integrals;74
4.8;2.8 Properties of Integrals;83
4.9;2.9 Relations between Integrals;99
4.10;2.10 Banach-Valued Contour Integrals and Cauchy Formulas;102
4.11;2.11 Notes;110
5;3 f- variation and p- variation; Inequalities for Integrals;115
5.1;3.1 f- variation;115
5.2;3.2 Interval Functions, f - variation, and p- variation;128
5.3;3.3 Elementary Properties;139
5.4;*3.4 A Necessary and Sufficient Condition for Integral Duality;160
5.5;3.5 Sufficient Conditions for Integrability;180
5.6;3.6 Love–Young Inequalities;185
5.7;3.7 Existence of Extended Riemann–Stieltjes Integrals;203
5.8;3.8 Convolution and Related Integral Transforms;208
5.9;3.9 Notes;223
6;4 Banach Algebras;227
6.1;4.1 Ideals and Normed Algebras;228
6.2;4.2 The Spectral Radius;231
6.3;4.3 Characters;234
6.4;4.4 Holomorphic Functions of Banach Algebra Elements;237
6.5;4.5 Complexification of Real Banach Algebras;243
6.6;4.6 A Substitution Rule for the Kolmogorov Integral;245
6.7;4.7 Notes;246
7;5 Derivatives and Analyticity in Normed Spaces;248
7.1;5.1 Polynomials and Power Series;250
7.2;5.2 Higher Order Derivatives and Taylor Series;262
7.3;5.3 Taylor’s Formulas;275
7.4;5.4 Tensor Products of Banach Spaces;280
7.5;5.5 Notes;281
8;6 Nemytskii Operators on Some Function Spaces;284
8.1;6.1 Overview;284
8.2;6.2 Remainders in Differentiability;285
8.3;6.3 Higher Order Differentiability and Analyticity;287
8.4;6.4 Autonomous Nemytskii Operators on Wp Spaces;304
8.5;6.5 Nemytskii Operators on Wp Spaces;311
8.6;6.6 Higher Order Differentiability and Analyticity on Wp Spaces;334
8.7;6.7 Notes;344
9;7 Nemytskii Operators on Lp Spaces;345
9.1;7.1 Acting, Boundedness, and Continuity Conditions;345
9.2;7.2 Hölder Properties;357
9.3;7.3 Differentiability;364
9.4;7.4 Higher Order Differentiability and Finite Taylor Series;378
9.5;7.5 Examples where NF Is Differentiable and F Is Not;388
9.6;7.6 Notes;400
10;8 Two-Function Composition;402
10.1;8.1 Overview; General Remarks;402
10.2;8.2 Differentiability of Two-Function Composition in General;403
10.3;8.3 Measure Space Domains;406
10.4;8.4 Spaces with Norms Stronger than Supremum Norms;410
10.5;8.5 Two-Function Composition on Wp Spaces;413
10.6;8.6 Notes;414
11;9 Product Integration;416
11.1;9.1 Multiplicative Interval Functions and f - Variation;417
11.2;9.2 Product Integrals for Real-Valued Interval Functions;424
11.3;9.3 Nonexistence for p > 2;433
11.4;9.4 Inequalities for Finite Products;434
11.5;9.5 The Product Integral;439
11.6;9.6 The Strict Product Integral;441
11.7;9.7 Commutative Banach Algebras;447
11.8;9.8 Integrals with Two Integrands;451
11.9;9.9 Duhamel’s Formula;460
11.10;9.10 Smoothness of the Product Integral Operator;466
11.11;9.11 Linear Integral Equations;483
11.12;9.12 Integral Equations for Banach-Space-Valued Functions;503
11.13;9.13 Notes;508
12;10 Nonlinear Differential and Integral Equations;513
12.1;10.1 Classical Picard Iteration;513
12.2;10.2 Picard’s Method in p- Variation Norm, 1 p < 2;514
12.3;10.3 Existence and Uniqueness;521
12.4;10.4 Boundedness and Convergence of Picard Iterates;535
12.5;10.5 Continuity of the Solution Mappings;544
12.6;10.6 Notes;557
13;11 Fourier Series;558
13.1;11.1 On the Order of Decrease of Fourier Coefficients;558
13.2;11.2 Uniform Convergence;561
13.3;11.3 Notes;575
14;12 Stochastic Processes and f - Variation;577
14.1;12.1 Processes with Regulated Sample Functions;577
14.2;12.2 Brownian Motion;581
14.3;12.3 Martingales;582
14.4;12.4 Gaussian Stochastic Processes;585
14.5;12.5 Markov Processes;606
14.6;12.6 Levy Processes;617
14.7;12.7 Empirical Processes;621
14.8;12.8 Differentiability of Operators on Processes;634
14.9;12.9 Integration;635
14.10;12.10 Notes;646
15;Appendix Nonatomic Measure Spaces;650
16;References;654
17;Subject Index;666
18;Author Index;670
19;Index of Notation;673