This book provides a concrete introduction to a number of topics in
harmonic analysis, accessible at the early graduate level or, in some
cases, at an upper undergraduate level. Necessary prerequisites to using
the text are rudiments of the Lebesgue measure and integration on the real
line. It begins with a thorough treatment of Fourier series on the circle
and their applications to approximation theory, probability, and plane
geometry (the isoperimetric theorem). Frequently, more than one proof is
offered for a given theorem to illustrate the multiplicity of approaches.
The second chapter treats the Fourier transform on Euclidean spaces,
especially the author's results in the three-dimensional piecewise smooth
case, which is distinct from the classical Gibbs–Wilbraham phenomenon of
one-dimensional Fourier analysis. The Poisson summation formula treated in
Chapter 3 provides an elegant connection between Fourier series on the
circle and Fourier transforms on the real line, culminating in Landau's
asymptotic formulas for lattice points on a large sphere.
Much of modern harmonic analysis is concerned with the behavior of
various linear operators on the Lebesgue spaces
$L^p(\mathbb{R}^n)$. Chapter 4 gives a gentle introduction to
these results, using the Riesz–Thorin theorem and the
Marcinkiewicz interpolation formula. One of the long-time users of
Fourier analysis is probability theory. In Chapter 5 the central limit
theorem, iterated log theorem, and Berry–Esseen theorems are
developed using the suitable Fourier-analytic tools.
The final chapter furnishes a gentle introduction to wavelet
theory, depending only on the $L_2$ theory of the Fourier
transform (the Plancherel theorem). The basic notions of scale and
location parameters demonstrate the flexibility of the wavelet
approach to harmonic analysis.
The text contains numerous examples and more than 200 exercises, each
located in close proximity to the related theoretical material.
Readership
Undergraduate and graduate students interested in Fourier
transform and harmonic analysis.
Download chapters as PDF