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Abstract

The purpose of the volume is to bring forward recent trends of research in hypercomplex analysis. The list of contributors includes first rate mathematicians and young researchers working on several different aspects in quaternionic and Clifford analysis. Besides original research papers, there are papers providing the state-of-the-art of a specific topic, sometimes containing interdisciplinary fields.

The intended audience includes researchers, PhD students and postgraduate students who are interested in the field and in connections between hypercomplex analysis and other disciplines, in particular mathematical analysis, mathematical physics, and algebra.

Contributors:

C. Bisi, F. Colombo, K. Coulembier, H. De Bie, S.-L. Eriksson, M. Fei, M. Ferreira, P. Franek, G. Gentili, R. Ghiloni, R.S. Kraußhar, R. Lávička, S. Li, M. Libine, M.E. Luna-Elizarrarás, M.A. Macías-Cedeño, M. Martin, H. Orelma, A. Perotti, I. Sabadini, M. Shapiro, P. Somberg, F. Sommen, C. Stoppato, D.C. Struppa, V. Tuček, A. Vajiac, M.B. Vajiac, F. Vlacci M.A. Macías-Cedeño, M. Martin, H. Orelma, A. Perotti, I. Sabadini, M. Shapiro, P. Somberg, F. Sommen, C. Stoppato, D.C. Struppa, V. Tuček, A. Vajiac, M.B. Vajiac, F. Vlacci.

Keywords

Functions of complex variables. --- Mathematical analysis. --- Clifford algebras. --- Geometric algebras --- Algebras, Linear --- 517.1 Mathematical analysis --- Mathematical analysis --- Complex variables --- Elliptic functions --- Functions of real variables --- Clifford algebras --- Analytic functions --- Differential equations, partial. --- Mathematical physics. --- Harmonic analysis. --- Numerical analysis. --- Partial Differential Equations. --- Mathematical Methods in Physics. --- Abstract Harmonic Analysis. --- Numerical Analysis. --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Banach algebras --- Calculus --- Mathematics --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis --- Physical mathematics --- Physics --- Partial differential equations --- Partial differential equations. --- Physics. --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics

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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.

Keywords

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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Keywords

Analytical spaces --- Harmonic analysis. Fourier analysis --- Partial differential equations --- Mathematical analysis --- Fourieranalyse --- differentiaalvergelijkingen

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This Briefs volume develops the theory of entire slice regular functions. It is the first self-contained, monographic work on the subject, offering all the necessary background information and detailed studies on several central topics, including estimates on the minimum modulus of regular functions, relations between Taylor coefficients and the growth of entire functions, density of their zeros, and the universality properties. The proofs presented here shed new light on the nature of the quaternionic setting and provide inspiration for further research directions. Also featuring an exhaustive reference list, the book offers a valuable resource for graduate students, postgraduate students and researchers in various areas of mathematical analysis, in particular hypercomplex analysis and approximation theory.

Keywords

Mathematics. --- Functions of complex variables. --- Functions of a Complex Variable. --- Mathematical analysis. --- 517.1 Mathematical analysis --- Mathematical analysis --- Complex variables --- Elliptic functions --- Functions of real variables

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This work contributes to the study of quaternionic linear operators. This study is a generalization of the complex case, but the noncommutative setting of quaternions shows several interesting new features, see e.g. the so-called S-spectrum and S-resolvent operators. In this work, we study de Branges spaces, namely the quaternionic counterparts of spaces of analytic functions (in a suitable sense) with some specific reproducing kernels, in the unit ball of quaternions or in the half space of quaternions with positive real parts. The spaces under consideration will be Hilbert or Pontryagin or Krein spaces. These spaces are closely related to operator models that are also discussed. The focus of this book is the notion of characteristic operator function of a bounded linear operator A with finite real part, and we address several questions like the study of J-contractive functions, where J is self-adjoint and unitary, and we also treat the inverse problem, namely to characterize which J-contractive functions are characteristic operator functions of an operator. In particular, we prove the counterpart of Potapov's factorization theorem in this framework. Besides other topics, we consider canonical differential equations in the setting of slice hyperholomorphic functions and we define the lossless inverse scattering problem. We also consider the inverse scattering problem associated with canonical differential equations. These equations provide a convenient unifying framework to discuss a number of questions pertaining, for example, to inverse scattering, non-linear partial differential equations and are studied in the last section of this book.

Keywords

Functional analysis. --- Operator theory. --- Functions of complex variables. --- Functional Analysis. --- Operator Theory. --- Functions of a Complex Variable. --- Complex variables --- Elliptic functions --- Functions of real variables --- Functional analysis --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Quaternions. --- Algebra, Universal --- Algebraic fields --- Curves --- Surfaces --- Numbers, Complex --- Vector analysis