Lecture |
Topics |
1 |
Review: Complex
numbers (esp. Euler's formula); periodic functions; functions on an interval;
functions on a circle; continuous functions; continuously differentiable
functions; Riemann integrable functions (or at least piecewise continuous
functions). Note that Lebesque integration and L^2 are not covered rigorously
in this course. |
2 |
Does
every function have a Fourier series? Formal computation of Fourier coefficients.
Inversion formula for trigonometric polynomials. Examples of Fourier series
(esp. Dirichlet kernel). |
3 |
Review
of convergence, uniform convergence. Do Fourier series converge back to
the original function? Injectivity of the Fourier transform for continuous
functions. |
4 |
Uniform
convergence for absolutely summable Fourier coefficients. Relationship between
differentiation and the Fourier transform. Uniform convergence for C^2 functions.
(Optional) Some foreshadowing of future convergence results. |
5 |
Convolutions of
continuous periodic functions: examples and basic properties. Connections
with Fourier coefficients. Connection between partial sums and the Dirichlet
kernel. |
6 |
Convolutions of
integrable periodic functions: approximation of integrable functions by
continuous ones. Approximation via convolution by good kernels. |
7 |
Badness of the Dirichlet
kernel; Gibbs' phenomenon. Cesaro means; Fejer kernel. Fejer's theorem.
Uniform approximation of continuous functions by trigonometric polynomials. |
8 |
Leeway |
9 |
Review of vector
spaces, inner product spaces, orthonormal sets, Cauchy-Schwarz inequality,
Pythagoras's theorem. Orthonormality of the Fourier basis. Bessel's inequality.
Best mean-square approximation by trigonometric polynomials. |
10 |
Mean-square convergence
of Fourier series for continuous functions. Mean-square convergence of Fourier
series for Riemann-integrable functions. Plancherel's theorem, Parseval's
theorem. Riemann-Lebesque lemma. |
11-12 |
Applications and
further properties of Fourier series, at instructor's discretion. Some suggestions:
Summation of 1/n^2; local convergence of Fourier series at smooth points;
smoothness of a function versus decay of Fourier coefficients; a continuous
function with divergent Fourier series; comparison of sine and cosine Fourier
series with exponential Fourier series; isoperimetic inequality; uniform
distribution of multiples of an irrational (Monte Carlo integration); a
continuous, nowhere differentiable function. |
13 |
Leeway/review |
14 |
Midterm. |
15 |
From Fourier series
to Fourier integrals - an informal discussion. Review of improper integrals.
Functions of moderate decrease. Functions of rapid decrease. Schwartz functions.
Definition of the Fourier transform. |
16 |
Basic algebraic
properties of the Fourier transform. Preservation of the Schwartz space. |
17 |
Fourier transform
of Gaussians. Gaussians as good kernels. |
18 |
Multiplication formula.
Fourier inversion formula. Bijectivity on Schwartz space. |
19 |
Fourier transform
and convolutions. Plancherel's theorem. Extension to functions of moderate
decrease. |
20-21 |
Integration on R^d;
Fourier transform on R^d; key properties. |
22-24 |
Applications to
PDE: heat equation; Laplace's equation. (Optional) The wave equation (in
1D or higher dimensions). |
25 |
Z_N. The finite
Fourier transform; key properties. |
26-28 |
Applications and
further properties of Fourier transforms, at instructor's discretion. Some
suggestions: The fast Fourier transform; fast multiplication; Heisenberg
uncertainty principle; Comparison of Fourier and Laplace transforms; The
Fourier-Bessel transform for radial functions; Poisson summation formula;
band-limited functions and the Shannon sampling theorem; linear transformations
and the Fourier transform on R^n; the Dirac delta function. |
29 |
Leeway/review. |
