School of Mathematics
Georgia Institute of Technology
Atlanta, GA 30332-0160
h e i l @ m a t h . g a t e c h . e d u
A Basis Theory Primer, Expanded Edition,
Birkhäuser/Springer, New York, 2011 (xxvi + 534 pp.).
From the back cover:
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 its 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.
- Part I develops the functional analysis that underlies most of the concepts
presented in the later parts of the text.
- Part II presents the abstract theory of bases and frames in Banach and Hilbert
spaces, including the classical topics of convergence, Schauder bases, biorthogonal
systems, and unconditional bases, followed by the more recent topics of Riesz bases
and frames in Hilbert spaces.
- Part III relates bases and frames to applied harmonic analysis, including sampling
theory, Gabor analysis, and wavelet theory.
- Part IV deals with classical harmonic analysis and Fourier series, emphasizing
the role played by bases, which is a different viewpoint from that taken in most
discussions of Fourier series.
Front and back covers.
Table of contents.
- See the book at:
Monatshefte für Mathematik, Vol. 166 (2012).
Quote: "The present book gives a wide perspective ...
The presentation is very reader-friendly and provides all necessary details."
Zentralblatt Math (2011).
Quote: "This book is a very comprehensive work ...
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
Quote: "It is one of those very few mathematical books that I can
read without additional difficulties ..."
Suggested Exercises for Self Study
- A Solutions Manual for Instructors is available for
instructors who register at the
This is copyrighted material, please do not distribute without permission.
Errata List (last updated October 2021).
Please email me with any typos or errors that you may find.
- Here are the very brief and limited
Original Notes that were the first incarnation of the Basis Primer.
my other books.