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Leon Ehrenpreis has been one of the leading mathematicians in the twentieth century. His contributions to the theory of partial differential equations were part of the golden era of PDEs, and led him to what is maybe his most important contribution,  the Fundamental Principle, which he announced in 1960, and fully demonstrated in 1970. His most recent work, on the other hand, focused on a novel and far reaching understanding of the Radon transform, and offered new insights in integral geometry. Leon Ehrenpreis died in 2010, and this volume collects writings in his honor by a cadre of distinguished mathematicians, many of which (Farkas, Kawai, Kuchment, Quinto) were his collaborators.

Ehrenpreis, Leon. --- Mathematicians. --- Mathematics -- History. --- Mathematics --- Mathematicians --- Civil & Environmental Engineering --- Physical Sciences & Mathematics --- Engineering & Applied Sciences --- Mathematics - General --- Operations Research --- History --- History. --- Math --- Ehrenpreis, Eliezer --- Mathematics. --- Fourier analysis. --- Partial differential equations. --- Functions of complex variables. --- Fourier Analysis. --- Several Complex Variables and Analytic Spaces. --- Partial Differential Equations. --- Complex variables --- Elliptic functions --- Functions of real variables --- Partial differential equations --- Analysis, Fourier --- Mathematical analysis --- Science --- Scientists --- Differential equations, partial.

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This book presents a functional calculus for n-tuples of not necessarily commuting linear operators. In particular, a functional calculus for quaternionic linear operators is developed. These calculi are based on a new theory of hyperholomorphicity for functions with values in a Clifford algebra: the so-called slice monogenic functions which are carefully described in the book. In the case of functions with values in the algebra of quaternions these functions are named slice regular functions.
 

Except for the appendix and the introduction all results are new and appear for the first time organized in a monograph. The material has been carefully prepared to be as self-contained as possible. The intended audience consists of researchers, graduate and postgraduate students interested in operator theory, spectral theory,  hypercomplex analysis, and mathematical physics.

Field theory (Physics). --- Function spaces. --- Functional analysis. --- Lp spaces. --- Noncommutative function spaces. --- Functional analysis --- Functions of complex variables --- Operator theory --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Non-commutative function spaces --- Functional calculus --- Spaces, Lp --- Spaces, Function --- Mathematics. --- Functions of complex variables. --- Operator theory. --- Operator Theory. --- Functional Analysis. --- Functions of a Complex Variable. --- Calculus of variations --- Functional equations --- Integral equations --- Function spaces --- Complex variables --- Elliptic functions --- Functions of real variables

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The theory of slice regular functions over quaternions is the central subject of the present volume. This recent theory has expanded rapidly, producing a variety of new results that have caught the attention of the international research community. At the same time, the theory has already developed sturdy foundations. The richness of the theory of the holomorphic functions of one complex variable and its wide variety of applications are a strong motivation for the study of its analogs in higher dimensions. In this respect, the four-dimensional case is particularly interesting due to its relevance in physics and its algebraic properties, as the quaternion forms the only associative real division algebra with a finite dimension n>2. Among other interesting function theories introduced in the quaternionic setting, that of (slice) regular functions shows particularly appealing features. For instance, this class of functions naturally includes polynomials and power series. The zero set of a slice regular function has an interesting structure, strictly linked to a multiplicative operation, and it allows the study of singularities. Integral representation formulas enrich the theory and they are a fundamental tool for one of the applications, the construction of a noncommutative functional calculus. The volume presents a state-of-the-art survey of the theory and a brief overview of its generalizations and applications. It is intended for graduate students and researchers in complex or hypercomplex analysis and geometry, function theory, and functional analysis in general.

Algebra. --- Mathematics. --- Polynomials. --- Functions, Quaternion --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Functional analysis. --- Functions of complex variables. --- Sequences (Mathematics) --- Mathematical sequences --- Numerical sequences --- Complex variables --- Functional calculus --- Math --- Sequences (Mathematics). --- Functions of a Complex Variable. --- Sequences, Series, Summability. --- Functional Analysis. --- Elliptic functions --- Functions of real variables --- Calculus of variations --- Functional equations --- Integral equations --- Science --- Algebra

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This book gathers contributions written by Daniel Alpay’s friends and collaborators. Several of the papers were presented at the International Conference on Complex Analysis and Operator Theory held in honor of Professor Alpay’s 60th birthday at Chapman University in November 2016. The main topics covered are complex analysis, operator theory and other areas of mathematics close to Alpay’s primary research interests. The book is recommended for mathematicians from the graduate level on, working in various areas of mathematical analysis, operator theory, infinite dimensional analysis, linear systems, and stochastic processes.

Mathematics. --- Functional analysis. --- Functions of complex variables. --- Operator theory. --- Functions of a Complex Variable. --- Functional Analysis. --- Operator Theory. --- Mathematical analysis. --- Complex variables --- Elliptic functions --- Functions of real variables --- 517.1 Mathematical analysis --- Mathematical analysis --- Functional analysis --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations

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The purpose of this book is to develop the foundations of the theory of holomorphicity on the ring of bicomplex numbers. Accordingly, the main focus is on expressing the similarities with, and differences from, the classical theory of one complex variable. The result is an elementary yet comprehensive introduction to the algebra, geometry and analysis of bicomplex numbers. Around the middle of the nineteenth century, several mathematicians (the best known being Sir William Hamilton and Arthur Cayley) became interested in studying number systems that extended the field of complex numbers. Hamilton famously introduced the quaternions, a skew field in real-dimension four, while almost simultaneously James Cockle introduced a commutative four-dimensional real algebra, which was rediscovered in 1892 by Corrado Segre, who referred to his elements as bicomplex numbers. The advantages of commutativity were accompanied by the introduction of zero divisors, something that for a while dampened interest in this subject. In recent years, due largely to the work of G.B. Price, there has been a resurgence of interest in the study of these numbers and, more importantly, in the study of functions defined on the ring of bicomplex numbers, which mimic the behavior of holomorphic functions of a complex variable. While the algebra of bicomplex numbers is a four-dimensional real algebra, it is useful to think of it as a “complexification” of the field of complex numbers; from this perspective, the bicomplex algebra possesses the properties of a one-dimensional theory inside four real dimensions. Its rich analysis and innovative geometry provide new ideas and potential applications in relativity and quantum mechanics alike. The book will appeal to researchers in the fields of complex, hypercomplex and functional analysis, as well as undergraduate and graduate students with an interest in one- or multidimensional complex analysis.

Calculus --- Mathematics --- Physical Sciences & Mathematics --- Holomorphic functions. --- Numbers, Complex. --- Complex numbers --- Imaginary quantities --- Quantities, Imaginary --- Functions, Holomorphic --- Mathematics. --- Functions of complex variables. --- Mathematical physics. --- Functions of a Complex Variable. --- Several Complex Variables and Analytic Spaces. --- Mathematical Applications in the Physical Sciences. --- Algebra, Universal --- Quaternions --- Vector analysis --- Functions of several complex variables --- Differential equations, partial. --- Partial differential equations --- Complex variables --- Elliptic functions --- Functions of real variables --- Physical mathematics --- Physics