Reading Course in Fourier Analysis -- Spring 1997
The
beginnings of the field of Fourier Analysis can be found in a book of
Joseph Fourier from the early 1800's devoted to the mathematical study of
heat conduction. To study the flow of heat through an object or a
volume of air (possibly containing one or more heat sources, possibly with
insulation at the boundaries, etc.), Fourier used the hypothesis that
functions on a finite closed interval, say [-pi,pi], can be represented
as an infinite linear combination of periodic sine and cosine functions
(now called Fourier series).
Fourier gave formulas for computing the coefficients in such expansions
and applied his methods to many physical problems with great success.
Nevertheless, he was vague (at best) in saying why expansions
would exist and in proving convergence. As a result, many
mathematicians of the 19th and 20th centuries considered the conditions
under which a Fourier series will actually converge (pointwise or
uniformly), in order to understand better the scope of applicability
of Fourier's methods. Indeed, the desire to understand the convergence
properties of these series was a major contributing factor in the greater
understanding of the real number system and the fundamentals of the theory of
analysis achieved during this period.
Note that if a function can be represented as a Fourier
series on the real line, then it must be a periodic function
with period 2 pi. In physical terms, a periodic function is a combination
of sinusoidal functions with a discrete spectrum of frequencies.
The Fourier coefficients tell ``how much'' of each frequency
is present in in the function.
As we will see, for non-periodic functions on the real line,
there is also Fourier transform that plays
the same role. The Fourier transform is a basic tool in signal processing,
electrical engineering, parts of optics, physics, etc.
This reading course was organized as follows:
- Fourier Series (about 6 weeks):
- Formal Fourier series computations,
- Pointwise convergence theorems via the Dirichlet kernel
function,
- Some L2 theory,
- Uniform convergence and the Gibbs phenomenon,
- Application to boundary value problems for the heat equation.
- Fourier Transforms (about 8 weeks):
- Motivation for
the Fourier transform,
- Lots of examples,
- Fourier Inversion,
- Convolution of functions and its intuitive meaning,
- The Nyquist Sampling
Theorem,
- Applications to signal processing (various types
of filters),
- The Discrete and Fast Fourier Transforms (algorithms
for computing approximations to the Fourier Transform by computer).
Text: Walker, Fourier Analysis
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