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Books on Fourier Analysis
There are many good textbooks in Fourier Analysis. I will list some of
them with my comments.
H. Dym and H. P. McKean: Fourier Series and Integrals, Academic
Press, 1972
This book contains numerous applications of Fourier analysis. Strongly recommended
for anyone who is interested in applications and wants to deepen their understanding
of Fourier analysis. It also includes a nice description of Lebesgue integration
and group theory
T. W. Körner: Fourier Analysis, Cambridge University
Press, 1988
This is a monumental work on Fourier analysis, consisting of a bunch of
interrelated essays. Read one section per day! You will gain a lot. Highly
recommended.
J. S. Walker: Fourier Analysis, Oxford University Press,
1988
A well-written and solid book on Fourier analysis with applications on optics,
computer-aided tomography, spherical harmonics, etc.
G. B. Folland: Fourier Analysis and Its Applications, Brooks/Cole
Publishing Co., 1992
An introductory but extremely well-written textbook on Fourier analysis.
Contains chapters on special functions, generalized functions (distributions),
and Greens functions. Applications are mainly for differential equations.
Expensive but worth buying it.
J. M. Ash (ed.): Studies in Harmonic Analysis, Mathematical
Association of America, 1976
This is a collection of conference talks by the authorities held in Chicago
in 1975. Most of the chapters are as if these authorities are directly
talking to you in a friendly manner about the essence of the ideas in harmonic
analysis without much detailed proofs. Contains really deep mathematics.
S. G. Krantz: A Panorama of Harmonic Analysis, Mathematical
Association of America, 1999
This book gives a historical perspective of harmonic analysis ranging from
classical to modern, from elementary to advanced. One can see how subtle
it is to sum multiple Fourier series. This also includes short description
on wavelets. Highly recommended.
E. M. Stein and G. Weiss: Introduction to Fourier Analysis on
Euclidean Spaces, Princeton University Press, 1971
A classic of the multidimensional Fourier analysis. Includes detailed discussions
on the invariance properties of Fourier transform.
A. Zygmund: Trigonometric Series (2nd Ed., Volume I &
II combined), Cambridge University Press, 1959
An ultimate bible on Fourier series and integrals for hard analysts.
This is basically a dictionary. Almost no applications are treated here.
R. N. Bracewell: The Fourier Transform and Its Applications
(2nd Ed., Revised), MacGraw-Hill, 1986
Another bible for engineers. Contains an excellent pictorial dictionary
of many functions and their Fourier transforms.
G. P. Tolstov: Fourier Series, Dover, 1972
The most cost effective book (about $12). Very well written. Highly recommended.
G. H. Hardy and W. W. Rogosinski: Fourier Series, Dover,
1999
This is a prelude to Zygmund's book. Spirit of pure mathematics. No applications
included. Economical ($7).
W. L. Briggs and V. E. Henson: The DFT: An Owner's Manual for
the Discrete Fourier Transform, SIAM 1995
This is a very useful book on DFT. Includes many practical applications,
such as tomography, seismic migrations, difference equation solvers. Detailed
analysis on the error of the DFT. A nice book to keep on your desk.
A. Terras: Fourier Analysis on Finite Groups and Applications
, Cambridge University Press, 1999
Another type of Fourier analysis. A more detailed version of the first half
of Chapter 4 of Dym and McKean plus many more examples and applications of
that aspect of Fourier analysis.