Preface
While such subjects as number theory and probability theory are commonly offered
to undergraduates, it seems that Fourier analysis is rarely found, which is shocking
when one considers the value of this subject not just within mathematics but also
in the physical sciences and engineering. The author hopes that this book will
encourage the view that Fourier analysis can be fruitfully presented not just to
undergraduates, but even to younger undergraduates with no more experience than
three or four terms of calculus. Such students will find a gentle introduction to the
art of writing proofs and will be better prepared for advanced calculus and complex
variables.
A student who has taken a course in advanced calculus may wonder what can be
done with that machinery. The answer is: harmonic analysis (among other things).
Paul Halmos is reported to have said words to the effect that the tragedy of Fourier
series is that they were invented (in 1807) before convergence. The wonderful
thing is that analysts such as Cauchy, Dirichlet, Riemann, and Weierstrass were
motivated to develop the foundations of real analysis in order to make sense of
Fourier series. In particular, Riemann defined his integral in order to provide a
more rigorous basis for the discussion of Fourier series.
This book could be used for a capstone course of an undergraduate program
or for beginning graduate students as a way to motivate the study of the Lebesgue
integral. Since it is hoped that this book will be useful at a wide range of levels, it
contains far more material than would ever be used in a single one term course. The
author will be happy to provide suggestions adjusted to the instuctor’s purpose.
We study Fourier analysis in three important settings. First we consider the
Discrete Fourier Transform, which has to do with the use of roots of unity to de-
scribe periodic sequences. The results in this setting are easily obtained, and they
form a framework for our endeavors in the more difficult subsequent settings. The
point is that in the discrete setting there is no issue of convergence, but with Fourier
Series we discover that convergence is a delicate matter. With Fourier Transforms
ix
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