Strange Functions in Real Analysis (Pure & Applied Mathematics) - Hardcover

Kharazishvili, Alexander

Hardcover

ISBN 10: 1584885823 ISBN 13: 9781584885825

Publisher: Chapman and Hall/CRC, 2005

Synopsis

Weierstrass and Blancmange nowhere differentiable functions, Lebesgue integrable functions with everywhere divergent Fourier series, and various nonintegrable Lebesgue measurable functions. While dubbed strange or "pathological," these functions are ubiquitous throughout mathematics and play an important role in analysis, not only as counterexamples of seemingly true and natural statements, but also to stimulate and inspire the further development of real analysis. Strange Functions in Real Analysis explores a number of important examples and constructions of pathological functions. After introducing the basic concepts, the author begins with Cantor and Peano-type functions, then moves to functions whose constructions require essentially noneffective methods. These include functions without the Baire property, functions associated with a Hamel basis of the real line, and Sierpinski-Zygmund functions that are discontinuous on each subset of the real line having the cardinality continuum. Finally, he considers examples of functions whose existence cannot be established without the help of additional set-theoretical axioms and demonstrates that their existence follows from certain set-theoretical hypotheses, such as the Continuum Hypothesis.

About this title

Review

"This book is devoted to strange (or singular) objects in various fields of mathematics, or, in other words, its' monsters and freaks'....The author is well-known due to his studies in this field and the book is based on his course of lectures given at the Institute of Applied Mathematics of Tbilisi State University in 1998-1999 academic year. One can see that this book is interesting for a wide circle of mathematicians who work in analysis and close branches of mathematics, as specialists as well as beginners. The book is written with clear and understandable language and is accessible even for graduate students." ---Zentralblatt fur Mathematik, 2000

Synopsis

Weierstrass and Blancmange nowhere differentiable functions, Lebesgue integrable functions with everywhere divergent Fourier series, and various nonintegrable Lebesgue measurable functions. While dubbed strange or 'pathological', these functions are ubiquitous throughout mathematics and play an important role in analysis, not only as counterexamples of seemingly true and natural statements, but also to stimulate and inspire the further development of real analysis. "Strange Functions in Real Analysis" explores a number of important examples and constructions of pathological functions.After introducing the basic concepts, the author begins with Cantor and Peano-type functions, then moves to functions whose constructions require essentially noneffective methods. These include functions without the Baire property, functions associated with a Hamel basis of the real line, and Sierpinski-Zygmund functions that are discontinuous on each subset of the real line having the cardinality continuum.

Finally, he considers examples of functions whose existence cannot be established without the help of additional set-theoretical axioms and demonstrates that their existence follows from certain set-theoretical hypotheses, such as the Continuum Hypothesis.

Edition details

• Publisher: Chapman and Hall/CRC
• Publication date: 2005
• Language: English
• ISBN 10: 1584885823
• ISBN 13: 9781584885825
• Binding: Hardcover
• Edition number: 2
• Number of pages: 432