Homeomorphisms in analysis / Casper Goffman, Togo Nishiura, Daniel Waterman.

Format:

Book

Author/Creator:

Goffman, Casper, 1913- author.

Nishiura, Togo, 1931- author.

Waterman, Daniel, author.

Series:

Mathematical surveys and monographs ; no. 54.

Mathematical surveys and monographs, 0076-5376 ; volume 54

Language:

English

Subjects (All):

Homeomorphisms.

Mathematical analysis.

Physical Description:

1 online resource (235 p.)

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [1997]

Language Note:

English

Summary:

This work 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 ]*w functions, the Blumberg theorem, bounded variation in the sense of Cesari, and various theorems on Fourier series and generalized bounded variation of a function.

Contents:

""Contents""; ""Preface""; ""The one dimensional case""; ""Mappings and measures on R[sup(n)]""; ""Fourier series""; ""Part 1. The One Dimensional Case""; ""Chapter 1. Subsets of R""; ""1.1. Equivalence classes""; ""1.2. Lebesgue equivalence of sets""; ""1.3. Density topology""; ""1.4. The Zahorski classes""; ""Chapter 2. Baire Class 1""; ""2.1. Characterization""; ""2.2. Absolutely measurable functions""; ""2.3. Example""; ""Chapter 3. Differentiability Classes""; ""3.1. Continuous functions of bounded variation""; ""3.2. Continuously differentiable functions""

""3.3. The class C[sup(n)][0,1]""""3.4. Remarks""; ""Chapter 4. The Derivative Function""; ""4.1. Properties of derivatives""; ""4.2. Characterization of the derivative""; ""4.3. Proof of Maximoff's theorem""; ""4.4. Approximate derivatives""; ""4.5. Remarks""; ""Part 2. Mappings and Measures on R[sup(n)]""; ""Chapter 5. Bi-Lipschitzian Homeomorphisms""; ""5.1. Lebesgue measurability""; ""5.2. Length of nonparametric curves""; ""5.3. Nonparametric area""; ""5.4. Invariance under self-homeomorphisms""; ""5.5. Invariance of approximately continuous functions""; ""5.6. Remarks""

""Chapter 6. Approximation by Homeomorphisms""""6.1. Background""; ""6.2. Approximations by homeomorphisms of one-to-one maps""; ""6.3. Extensions of homeomorphisms""; ""6.4. Measurable one-to-one maps""; ""Chapter 7. Measures on R[sup(n)]""; ""7.1. Preliminaries""; ""7.2. The one variable case""; ""7.3. Constructions of deformations""; ""7.4. Deformation theorem""; ""7.5. Remarks""; ""Chapter 8. Blumberg's Theorem""; ""8.1. Blumberg's theorem for metric spaces""; ""8.2. Non-Blumberg Baire spaces""; ""8.3. Homeomorphism analogues""; ""Part 3. Fourier Series""

""Chapter 9. Improving the Behavior of Fourier Series""""9.1. Preliminaries""; ""9.2. Uniform convergence""; ""9.3. Conjugate functions and the Pál-Bohr theorem""; ""9.4. Absolute convergence""; ""Chapter 10. Preservation of Convergence of Fourier Series""; ""10.1. Tests for pointwise and uniform convergence""; ""10.2. Fourier series of regulated functions""; ""10.3. Uniform convergence of Fourier series""; ""Chapter 11. Fourier Series of Integrable Functions""; ""11.1. Absolutely measurable functions""; ""11.2. Convergence of Fourier series after change of variable""

""11.3. Functions of generalized bounded variation""""11.4. Preservation of the order of magnitude of Fourier coefficients""; ""Appendix A. Supplementary Material""; ""Sets, Functions and Measures""; ""A.1. Baire, Borel and Lebesgue""; ""A.2. Lipschitzian functions""; ""A.3. Bounded variation""; ""Approximate Continuity""; ""A.4. Density topology""; ""A.5. Approximately continuous maps into metric spaces""; ""Hausdorff Measure and Packing""; ""A.6. Hausdorff dimension""; ""A.7. Hausdorff packing""; ""Nonparametric Length and Area""; ""A.8. Nonparametric length""; ""A.9. Schwarz's example""

""A.10. Lebesgue's lower semicontinuous area""

Notes:

Description based upon print version of record.

Includes bibliographical references (pages207 -211) and index.

Description based on print version record.