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Strong Boundary Values, Analytic Functionals, and Nonlinear Paley-Wiener Theory
Jean-Pierre Rosay, University of Wisconsin, Madison, WI, and Edgar Lee Stout, University of Washington, Seattle, WA
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Memoirs of the American Mathematical Society
2001; 94 pp; softcover
Volume: 153
ISBN-10: 0-8218-2712-X
ISBN-13: 978-0-8218-2712-3
List Price: US$51
Individual Members: US$30.60
Institutional Members: US$40.80
Order Code: MEMO/153/725
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We introduce a notion of boundary values for functions along real analytic boundaries, without any restriction on the growth of the functions. Our definition does not depend on having the functions satisfy a differential equation, but it covers the classical case of non-characteristic boundaries. These boundary values are analytic functionals or, in the local setting, hyperfunctions. We give a characterization of nonconvex carriers of analytic functionals, in the spirit of the Paley-Wiener-Martineau theory for convex carriers. Our treatment gives a new approach even to the classical Paley-Wiener theorem. The result applies to the study of analytic families of analytic functionals. The paper is mostly self contained. It starts with an exposition of the basic theory of analytic functionals and hyperfunctions, always using the most direct arguments that we have found. Detailed examples are discussed.

Readership

Graduate students and research mathematicians interested in functional analysis, several complex variables, analytic spaces, and differential equations.

Table of Contents

  • Introduction
  • Preliminaries on analytic functionals and hyperfunctions
  • Appendix on good compact sets
  • Analytic functionals as boundary values
  • Nonlinear Paley-Wiener theory
  • Strong boundary values
  • Strong boundary values for the solutions of certain partial differential equations
  • Comparison with other notions of boundary values
  • Boundary values via cousin decompositions
  • The Schwarz reflection principle
  • References
  • Index of notions
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