Measure, Integration & Real Analysis, 2021a

Chapter 11
Fourier Analysis
This chapter uses Hilbert space theory to motivate the introduction of Fourier coefficients
and Fourier series. The classical setting applies these concepts to functions
defined on bounded intervals of the real line. However, the theory becomes easier and
cleaner when we instead use a modern approach by considering functions defined on
the unit circle of the complex plane.
The first section of this chapter shows how consideration of Fourier series leads us
to harmonic functions and a solution to the Dirichlet problem. In the second section
of this chapter, convolution becomes a major tool for the L p theory.
The third section of this chapter changes the context to functions defined on the
real line. Many of the techniques introduced in the first two sections of the chapter
transfer easily to provide results about the Fourier transform on the real line. The
highlights of our treatment of the Fourier transform are the Fourier Inversion Formula
and the extension of the Fourier transform to a unitary operator on L 2 (R).
The vast field of Fourier analysis cannot be completely covered in a single chapter.
Thus this chapter gives readers just a taste of the subject. Readers who go on from
this chapter to one of the many book-length treatments of Fourier analysis will then
already be familiar with the terminology and techniques of the subject.
The Giza pyramids, near where the Battle of Pyramids took place in 1798 during
Napoleon’s invasion of Egypt. Joseph Fourier (1768–1830) was one of the scientific
advisors to Napoleon in Egypt. While in Egypt as part of Napoleon’s invading force,
Fourier began thinking about the mathematical theory of heat propagation, which
eventually led to what we now call Fourier series and the Fourier transform.
CC-BY-SA Ricardo Liberato
Measure, Integration & Real Analysis, by Sheldon Axler
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