Harmonic Analysis is an important tool that
plays a vital role in many areas of mathematics as well as
applications. It studies functions by decomposing them into components
that are special functions. A prime example is decomposing a periodic
function into a linear combination of sines and cosines. The subject
is vast, and this book covers only the selection of topics that was
dealt with in the course given at the Courant Institute in 2000 and
2019. These include standard topics like Fourier series and Fourier
transforms of functions, as well as issues of convergence of Abel,
Feier, and Poisson sums. At a slightly more advanced level the book
studies convolutions with singular integrals, fractional derivatives,
Sobolev spaces, embedding theorems, Hardy spaces, and
BMO. Applications to elliptic partial differential equations and
prediction theory are explored. Some space is devoted to harmonic
analysis on compact non-Abelian groups and their representations,
including some details about two groups: the permutation group and
SO(3).
The text contains exercises at the end of most chapters and is
suitable for advanced undergraduate students as well as first- or
second-year graduate students specializing in the areas of analysis,
PDE, probability or applied mathematics.
Readership
Undergraduate and graduate students interested in
harmonic analysis.