Concrete Functional Calculus

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.

• Composition operator
• Integration
• Interval functions
• Nemytskii operator
• Nonlinear integral equations
• Product integral
• Riemann-Stieltjes integral
• p-variation

From the reviews:

“This monograph is a thorough and masterful work on non-linear analysis designed to be read and studied by graduate students and professional mathematical researchers. The overall perspective and choice of material is highly novel and original. … It is a unique account of some key areas of modern analysis which will surely turn out to be invaluable for many researchers in this and related areas.” (David Applebaum, The Mathematical Gazette, Vol. 98 (541), March, 2014)

“The present monograph is quite extensive and interesting. It is divided into twelve chapters on different topics on Functional calculus and an appendix on non-atomic measure spaces. … The book has many historical comments and remarks which clarify the developments of the theory. It has also an extensive bibliography with 258 references. … will be very useful for all interested readers in Real-Functional Analysis and Probability.” (Francisco L. Hernandez, The European Mathematical Society, January, 2012)

“The monograph under review aims at analyzing properties such as Hölder continuity, differentiability and analyticity of various types of nonlinear operators which arises in the study of differential and integral equations and in applications to problems of statistics and probability. … this is an interesting book which contains a lot of material.” (Massimo Lanza de Cristoforis, Mathematical Reviews, Issue 2012 e)

• Technology, Department of Mathematics, Massachusetts Institute of, Cambridge, USA

  R. M. Dudley

• Institute of Mathematics and Informatics, Vilnius, Lithuania

  R. Norvaiša

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 Norvaiša is a principal researcher at the Institute of Mathematics and Informatics in Lithuania. Dudley and Norvaiša 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".

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