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Basics of Functional Analysis with Bicomplex Scalars, and Bicomplex Schur Analysis

  • Book
  • © 2014

Overview

Authors:
  1. Daniel Alpay

    1. Department of Mathematics, Ben-Gurion University of the Negev, Beer Sheva, Israel

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  2. Maria Elena Luna-Elizarrarás

    1. Departamento de Matemáticas, ESFM-IPN, Mexico D.F., Mexico

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  3. Michael Shapiro

    1. Departamento de Matemáticas, ESFM-IPN, Mexico D.F., Mexico

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  4. Daniele C. Struppa

    1. Schmid college of Science and Technology, Chapman University, Orange, USA

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  • Offers a self-contained introduction to functional analysis with bicomplex scalars and will serve for many subsequent developments similar to those of classic functional analysis
  • Only book that introduces a new hyperbolic valued norm on bicomplex modules which is crucial in obtaining some of the most important results in the book
  • First book in which Schur analysis is introduced in the bicomplex context
  • Includes supplementary material: sn.pub/extras

Part of the book series: SpringerBriefs in Mathematics (BRIEFSMATH)

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Table of contents (6 chapters)

  1. Front Matter

    Pages i-xv
    Download chapter PDF

  2. Bicomplex and Hyperbolic Numbers

    • Daniel Alpay, Maria Elena Luna-Elizarrarás, Michael Shapiro, Daniele C. Struppa
    Pages 1-17

  3. Bicomplex Functions and Matrices

    • Daniel Alpay, Maria Elena Luna-Elizarrarás, Michael Shapiro, Daniele C. Struppa
    Pages 19-30

  4. \(\mathbb B\mathbb C\)-Modules

    • Daniel Alpay, Maria Elena Luna-Elizarrarás, Michael Shapiro, Daniele C. Struppa
    Pages 31-36

  5. Norms and Inner Products on \(\mathbb B\mathbb C\)-Modules

    • Daniel Alpay, Maria Elena Luna-Elizarrarás, Michael Shapiro, Daniele C. Struppa
    Pages 37-61

  6. Linear Functionals and Linear Operators on \({\mathbb {B}}{\mathbb {C}}\)-Modules

    • Daniel Alpay, Maria Elena Luna-Elizarrarás, Michael Shapiro, Daniele C. Struppa
    Pages 63-77

  7. Schur Analysis

    • Daniel Alpay, Maria Elena Luna-Elizarrarás, Michael Shapiro, Daniele C. Struppa
    Pages 79-95
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Keywords

About this book

This book provides the foundations for a rigorous theory of functional analysis with bicomplex scalars. It begins with a detailed study of bicomplex and hyperbolic numbers and then defines the notion of bicomplex modules. After introducing a number of norms and inner products on such modules (some of which appear in this volume for the first time), the authors develop the theory of linear functionals and linear operators on bicomplex modules. All of this may serve for many different developments, just like the usual functional analysis with complex scalars and in this book it serves as the foundational material for the construction and study of a bicomplex version of the well known Schur analysis.

Authors and Affiliations

  • Department of Mathematics, Ben-Gurion University of the Negev, Beer Sheva, Israel

    Daniel Alpay

  • Departamento de Matemáticas, ESFM-IPN, Mexico D.F., Mexico

    Maria Elena Luna-Elizarrarás, Michael Shapiro

  • Schmid college of Science and Technology, Chapman University, Orange, USA

    Daniele C. Struppa

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