Fourier Analysis

COURSE OBJECTIVE
At the end of this course the student is able to:
a) Calculate the Fourier series of a given Riemann-integrable function
b) Determine the pointwise and prove the mean-square convergence of a Fourier series
c) Determine good kernels
d) Apply Fourier series theory to Cesàro and Abel summability
e) Calculate the Fourier transfom on the real line
f) Apply the Fourier transform to some PDE's

COURSE CONTENT
Topics that will treated are:
a) The genesis of Fourier Analysis, in particular the investigation of the wave equation
b) Basic Properties of Fourier Series (uniqueness, convolutions, Dirichlet and Poisson kernels)
c) Convergence of Fourier Series (pointwise, mean-square)
d) Cesàro and Abel Summability
e) Some applications of Fourier Series
f) The Fourier transform on the real line (definition, inversion, Plancherel formula)
g) Applications of the Fourier transform to some partial differential equations

TEACHING METHODS
Lectures (1x2 hours per week) and Tutorials (1x2 hours per week).

TYPE OF ASSESSMENT
There are hand-in exercises with grade H, a midterm with grade M and a final exam with grade F. Let A=.5(M+F) and B=.1 H+.9 A. To pass the course the student must have A>=.5 and B>=5.5. The final grade is then B. There is one resit opportunity for the full course. The grade for the homework does not count toward the grade of resit.

ENTRY REQUIREMENTS
Single variable calculus, Multivariable calculus, Linear algebra, and Mathematical analysis.