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Nonstandard Methods in Commutative Harmonic Analysis
E. I. Gordon, Nizhny Novgorod State University, Russia
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Translations of Mathematical Monographs
1997; 166 pp; hardcover
Volume: 164
ISBN-10: 0-8218-0419-7
ISBN-13: 978-0-8218-0419-3
List Price: US$90 Member Price: US$72
Order Code: MMONO/164

This monograph discusses nonstandard analysis (NSA) and its applications to harmonic analysis on locally compact Abelian (LCA) groups. A new notion of approximation of topological groups by finite groups is introduced and investigated. Based on this notion, new results are obtained on convergence of finite Fourier transformations (FT) to the FT on an LCA group. These results, formulated in standard terms in the Introduction, are proved by means of NSA. The book also includes new results on the theory of relatively standard elements and extensions of results of $$S$$-integrable liftings in Loeb measure spaces to the case of $$\sigma$$-finite Loeb measures. Basic concepts of NSA are included.

Graduate students and research mathematicians interested in mathematical analysis and mathematical and functional analysis, theoretical physicists and computer scientists.

Reviews

"Perfectly suited for running a study group among a number of professional mathematicians and senior students willing to start from scratch and to develop a working knowledge of nonstandard analysis taking them right to the cutting edge of current research."

-- Mathematical Reviews

• Introduction
• Basic concepts of nonstandard analysis
• Nonstandard analysis of operators acting in spaces of measurable functions
• Nonstandard analysis of locally compact Abelian groups
• Bibliography