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Heil, Christopher

A Basis Theory Primer Expanded Edition (Gebundene Ausgabe)

Originaltitel: A Basis Theory Primer Reihe: Applied and Numerical Harmonic Analysis

Springer Basel AG, Birkh„user Boston, Dezember 2010

536 S. - Sprache: Englisch - Originalsprache: Englisch - 42 schwarz-weiße Abbildungen, 1 schwarz-weiße Tabellen - 246x166x43 mm

ISBN: 0817646868 EAN: 9780817646868

This textbook is a self-contained introduction to the abstract theory of bases and redundant frame expansions and their use in both applied and classical harmonic analysis. The four parts of the text take the reader from classical functional analysis and basis theory to modern time-frequency and wavelet theory. Extensive exercises complement the text and provide opportunities for learning-by-doing, making the text suitable for graduate-level courses. The self-contained presentation with clear proofs is accessible to graduate students, pure and applied mathematicians, and engineers interested in the mathematical underpinnings of applications.

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This textbook is a mathematical introduction to the theory of bases in Banach spaces, also weaving in the theory of bases in Hilbert spaces and generalizations of bases such as frames. A new impetus for the study of bases in Banach spaces has come from the rise of wavelet theory, and especially the theory of frames, which allow basis-like but possibly redundant expansions.This is a text to learn from, with detailed explanations of abstract material that are usually explained with the remark "clearly ..." Covering abstract material with a high degree of relevance to a wide variety of modern topics, the book is primarily aimed at mathematicians, but will also encourage engineers to venture into new directions. Exercises at the end of each chapter make the work suitable as a textbook for a variety of graduate-level courses on bases and frames.


ANHA Series Preface.- Preface.- General Notation.- Part I. A Primer on Functional Analysis .- Banach Spaces and Operator Theory.- Functional Analysis.- Part II. Bases and Frames.- Unconditional Convergence of Series in Banach and Hilbert Spaces.- Bases in Banach Spaces.- Biorthogonality, Minimality, and More About Bases.- Unconditional Bases in Banach Spaces.- Bessel Sequences and Bases in Hilbert Spaces.- Frames in Hilbert Spaces.- Part III. Bases and Frames in Applied Harmonic Analysis.- The Fourier Transform on the Real Line.- Sampling, Weighted Exponentials, and Translations.- Gabor Bases and Frames.- Wavelet Bases and Frames.- Part IV. Fourier Series.- Fourier Series.- Basic Properties of Fourier Series.- Part V. Appendices.- Lebesgue Measure and Integration.- Compact and Hilbert-Schmidt Operators.- Hints for Exercises.- Index of Symbols.- References.- Index.


"This book is a very comprehensive work dedicated to introducing graduate students or researchers in pure and applied mathematics as well as engineering to the foundations of basis expansions and to essential techniques for applications...Chris Heil's book explains many aspects of Fourier's legacy which go beyond harmonic analysis, including Banach space geometry, nonharmonic Fourier analysis, and the modern theory of frames. The exercises contained in the book make it a good fit for graduate courses on selected topics in functional analysis and applications...[T]he book completes a tour from the fundamentals of functional analysis to currently active research related to basis expansions...Many topics in Chris Heil's book are also covered [elsewhere] but the level of detail found here is greater."   -ZentralBlatt MATH"The amount of mathematics treated in the book is impressive...Several results exposed in Part 3 are related to hot topics in wavelet analysis. Personally I like this book. It is one of those very few mathematical books that I can read without additional difficulties arising from my limited capacity to remember facts and definitions."   -Mathematical Reviews

ISBN 0-817646-86-8, ISBN 0-81-764686-8, ISBN 0-81764-686-8, ISBN 0-817-64686-8, ISBN 0-8176-4686-8

ISBN 978-0-817646-86-8, ISBN 978-0-81-764686-8, ISBN 978-0-81764-686-8, ISBN 978-0-817-64686-8, ISBN 978-0-8176-4686-8

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