In Section 21 the basic theory of sectorial operators is developed, including examples and the concept of sectorial approximation In Section 22 we introduce some notation for certain spaces of holomorphic functions on sectors A functional calculus for sectorial operators is constructed in Section 23 along the lines of the abstract framework of Chapter 1 Fundamental properties like the composition rule are proved In Section 25 we give natural extensions of the functional calculus to larger function spaces in the case where the given operator is bounded and/or invertible In this way a panorama of functional calculi is developed In Section 26 some mixed topics are discussed, eg, adjoints and restrictions of sectorial operators and some fundamental boundedness and some first approximation results Section 27 contains a spectral mapping theorem

The Functional Calculus for Sectorial Operators

General Theory of Banach Algebra. By C.E. Rickart. Pp. 394. 79s. 1960. (Van Nostrand, London)

This chapter gives a brief survey of continuous time finance. We categorize diffusion models according to the nature of their volatility coefficient. Models whose volatility coefficient does not exhibit randomness are treated in Sect. 3.1. Models whose volatility coefficient follows a stochastic process are discussed in Sect. 3.2.

Continuous Time Finance

A partial overview of the theory of statistics with functional data

The purpose of this book is to develop the foundations of potential theory and rational dynamics on the Berkovich projective line over an arbitrary complete, algebraically closed non-Archimedean field. In addition to providing a concrete and 'elementary' introduction to Berkovich analytic spaces and to potential theory and rational iteration on the Berkovich line, the book contains applications to arithmetic geometry and arithmetic dynamics. A number of results in the book are new, and most have not previously appeared in book form. Three appendices - on analysis, R-trees, and Berkovich's general theory of analytic spaces - are included to make the book as self-contained as possible. The authors first give a detailed description of the topological structure of the Berkovich projective line and then introduce the Hsia kernel, the fundamental kernel for potential theory. Using the theory of metrized graphs, they define a Laplacian operator on the Berkovich line and construct theories of capacities, harmonic and subharmonic functions, and Green's functions, all of which are strikingly similar to their classical complex counterparts. After developing a theory of multiplicities for rational functions, they give applications to non-Archimedean dynamics, including local and global equidistribution theorems, fixed point theorems, and Berkovich space analogues of many fundamental results from the classical Fatou-Julia theory of rational iteration. They illustrate the theory with concrete examples and exposit Rivera-Letelier's results concerning rational dynamics over the field of p-adic complex numbers. They also establish Berkovich space versions of arithmetic results such as the Fekete-Szego theorem and Bilu's equidistribution theorem.

Potential Theory and Dynamics on the Berkovich Projective Line

Potential theory is the broad area of mathematical analysis encompassing such topics as harmonic and subharmonic functions, the Dirichlet problem, harmonic measure, Green's functions, potentials and capacity. This is an introduction to the subject suitable for beginning graduate students, concentrating on the important case of two dimensions. This permits a simpler treatment than other books, yet is still sufficient for a wide range of applications to complex analysis; these include Picard's theorem, the Phragmen–Lindelof principle, the Koebe one-quarter mapping theorem and a sharp quantitative form of Runge's theorem. In addition there is a chapter on connections with functional analysis and dynamical systems, which shows how the theory can be applied to other parts of mathematics, and gives a flavour of some recent research. Exercises are provided throughout, enabling the book to be used with advanced courses on complex analysis or potential theory.

Potential theory in the complex plane

The Dirichlet space is one of the three fundamental Hilbert spaces of holomorphic functions on the unit disk. It boasts a rich and beautiful theory, yet at the same time remains a source of challenging open problems and a subject of active mathematical research. This book is the first systematic account of the Dirichlet space, assembling results previously only found in scattered research articles, and improving upon many of the proofs. Topics treated include: the Douglas and Carleson formulas for the Dirichlet integral, reproducing kernels, boundary behaviour and capacity, zero sets and uniqueness sets, multipliers, interpolation, Carleson measures, composition operators, local Dirichlet spaces, shift-invariant subspaces, and cyclicity. Special features include a self-contained treatment of capacity, including the strong-type inequality. The book will be valuable to researchers in function theory, and with over 100 exercises it is also suitable for self-study by graduate students.

https://assets.cambridge.org/97811070/47525/copyright/9781107047525_copyright_info.pdf

A Primer on the Dirichlet Space

The Cramer–Wold theorem states that a Borel probability measure P on ℝd is uniquely determined by its one-dimensional projections. We prove a sharp form of this result, addressing the problem of how large a subset of these projections is really needed to determine P. We also consider extensions of our results to measures on a separable Hilbert space.

/pdf/a-sharp-form-of-the-cramer-wold-theorem-3peoa5463y.pdf

A Sharp Form of the Cramér-Wold Theorem

In this paper we provide conditions under which a distribution is determined by just one randomly chosen projection. Thenweapply our results to construct goodness-of-fit tests for the one and two-sample problems. We include some simulations as well as the application of our results to a real data set. Our results are valid for every separable Hilbert space.

/pdf/random-projections-and-goodness-of-fit-tests-in-infinite-2sfdte24fv.pdf

Thomas Ransford

Papers

Potential theory in the complex plane

A Primer on the Dirichlet Space

A Sharp Form of the Cramér-Wold Theorem

Random projections and goodness-of-fit tests in infinite-dimensional spaces

A Primer on the Dirichlet Space: Cyclicity