Concrete Functional Calculus

This book 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. In this book existence and uniqueness of solutions are proved under suitable assumptions for nonlinear integral equations with respect to possibly discontinuous functions having unbounded variation. Key features and topics: Extensive usage of p-variation of functions, and 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.

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.

Opens new possibilities to analysis of statistical functionals and gives an alternative (non-Ito) approach to stochastic calculus Contains new material about the existence and smoothnesss of several nonlinear operators acting between spaces of functions having bounded p-variation Will appeal to graduate students and researchers working on various aspects of calculus of non-smooth functions Includes supplementary material: sn.pub/extras

Auteur

Richard M. Dudley is a professor of mathematics at MIT. He has published over a hundred papers in peer-reviewed journals and two books. He was one of three lecturers in the 1982 St.-Flour Summer School in Probability, published in Springer's Lecture Notes in Mathematics series in 1984.

Rimas Norvaia is a principal researcher at the Institute of Mathematics and Informatics in Lithuania. Dudley and Norvaia have written one previous book together in 1999 for Springer's Lecture Notes in Mathematics series, entitled "Differentiability of Six Operators on Nonsmooth Functions and P-Variation".

Contenu
Preface.- 1 Introduction and Overview.- 2 Definitions and Basic Properties of Extended Riemann-Stieltjes Integrals.- 3 Phi-variation and p-variation; Inequalities for Integrals.- 4 Banach Algebras.- 5 Derivatives and Analyticity in Normed Spaces.- 6 Nemytskii Operators on Some Function Spaces.- 7 Nemytskii Oerators on Lp Spaces.- 8 Two-Function Composition.- 9 Product Integration.- 10 Nonlinear Differential and Integral Equations.- 11 Fourier Series.- 12 Stochastic Processes and Phi-Variation.- Appendix Nonatomic Measure Spaces.- References.- Subject Index.- Author Index.- Index of Notation.