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Homeomorphisms in Analysis
Casper Goffman, Purdue University, West Lafayette, IN, Togo Nishiura, Wayne State University, Detroit, MI, and Daniel Waterman, Syracuse University, Syracuse, NY
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Mathematical Surveys and Monographs
1997; 216 pp; hardcover
Volume: 54
ISBN-10: 0-8218-0614-9
ISBN-13: 978-0-8218-0614-2
List Price: US$84
Member Price: US$67.20
Order Code: SURV/54
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This book features the interplay of two main branches of mathematics: topology and real analysis. The material of the book is largely contained in the research publications of the authors and their students from the past 50 years. Parts of analysis are touched upon in a unique way, for example, Lebesgue measurability, Baire classes of functions, differentiability, \(C^n\) and \(C^{\infty}\) functions, the Blumberg theorem, bounded variation in the sense of Cesari, and various theorems on Fourier series and generalized bounded variation of a function.

Features:

  • Contains new results and complete proofs of some known results for the first time.
  • Demonstrates the wide applicability of certain basic notions and techniques in measure theory and set-theoretic topology.
  • Gives unified treatments of large bodies of research found in the literature.

Readership

Graduate students, research mathematicians and engineers interested in real functions.

Reviews

"[T]he book is an extensive survey on the results in analysis which concern homeomorphisms. The material is presented in a clear form. Proofs of longer theorems are usually broken down into lemmas. Many examples and comments further facilitate the reading. The text is rounded off with an appendix consisting of supplementary material, an extensive bibliography on the subject, and an index."

-- Mathematical Reviews

"The book is well written, packed with information and makes a novel contribution to the literature. Much of what is in the book is important material that is now for the first time readily accessible ... readers will appreciate the many comments that provide historical or motivational perspectives."

-- Professor Andrew Bruckner, University of California, Santa Barbara

"[This] is, overall, an excellent book, of a type which, in this reviewer's opinion, is sorely lacking in mathematics these days."

-- Bulletin of the London Mathematical Society

Table of Contents

Part 1. The One Dimensional Case
  • Subsets of \(\mathbb{R}\)
  • Baire class 1
  • Differentiability classes
  • The derivative function
Part 2. Mappings and Measures on \({\mathbb R}^n\)
  • Bi-Lipschitzian homeomorphisms
  • Approximation by homeomorphisms
  • Measures on \(\mathbb{R}^n\)
  • Blumberg's theorem
Part 3. Fourier Series
  • Improving the behavior of Fourier series
  • Preservation of convergence of Fourier series
  • Fourier series of integrable functions
  • Supplementary material
  • Bibliography
  • Index
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