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A Basis Theory Primer

Expanded Edition

  • Textbook
  • © 2011


  • Unique book in the literature
  • A clear and accessible text with detailed explanations of abstract material
  • Suitable for classroom use or independent study
  • Covers abstract material with a high degree of relevance to a wide range of modern topics
  • Written for a broad audience of graduate students, pure and applied mathematicians as well as engineers
  • Includes extensive exercises at the end of each section
  • Separate solutions manual available for instructors upon request
  • Includes supplementary material:
  • Request lecturer material:

Part of the book series: Applied and Numerical Harmonic Analysis (ANHA)

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Table of contents (16 chapters)

  1. A Primer on Functional Analysis

  2. Bases and Frames

  3. Bases and Frames in Applied Harmonic Analysis

  4. Fourier Series

  5. Appendices


About this book

The classical subject of bases in Banach spaces has taken on a new life in the modern development of applied harmonic analysis. 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. No other text develops the ties between classical basis theory and its modern uses in applied harmonic analysis.


From the reviews:

“The present book gives a wide perspective, preparing the functional analytic ground … and also discussing in great detail the relevant features of bases in Banach spaces, unconditional bases, frames, and their role in the context of Applied Harmonic Analysis. … the book is ideally suited for self-study, but also as a text book from which different courses can be compiled. The presentation is very reader-friendly and provides all necessary details.” (H. G. Feichtinger, Monatshefte für Mathematik, Vol. 166 (3-4), June, 2012)

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. … The exercises contained in the book make it a good fit for graduate courses on selected topics in functional analysis and applications.” (Bernhard Bodmann, Zentralblatt MATH, Vol. 1227, 2012)

“The amount of mathematics treated in the book is impressive. … a handbook for a certain group of mathematicians to learn about the main tools of the theory of bases and frames for Banach and Hilbert spaces. … 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.” (Kazaros Kazarian, Mathematical Reviews, Issue 2012 b)

Authors and Affiliations

  • , School of Mathematics, Georgia Institute of Technology, Atlanta, USA

    Christopher Heil

Bibliographic Information

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