This course is intended to introduce basic concepts in the Fourier analysis, in particular, the Fourier series.
The course is followed by Applied Analysis II.
The main objective is to understand the mathematical treatment of Fourier series, which was originally introduced by Fourier himself for the purpose of solving the heat equation.
We study fundamental properties of Fourier series and its convergence, and applications to several fields in mathematics.
Students are expected to understand basic concepts in Fourier series, in particular, mathematical treatment of Fourier series.
We also focus on computations of Fourier series expansion of given functions and applications to differential equations.
Series of functions, Fourier series, Bessel's inequality, Riemann-Lebesgue lemma, Dirichlet kernel
|Intercultural skills||Communication skills||Specialist skills||Critical thinking skills||Practical and/or problem-solving skills|
Before coming to class, students should read the course schedule and check what topics will be covered.
Required learning should be completed outside of the classroom for preparation and review purposes.
|Course schedule||Required learning|
|Class 1||Fourier's idea and trigonometric series||Details will be provided during each class session|
|Class 2||Complex-valued functions and series of functions||Details will be provided during each class session|
|Class 3||Fourier series of a periodic function||Details will be provided during each class session|
|Class 4||Convergence theorem||Details will be provided during each class session|
|Class 5||Regularity of a function and the behavior of Fourier coefficients||Details will be provided during each class session|
|Class 6||Fourier series on intervals||Details will be provided during each class session|
|Class 7||Applications of Fourier series||Details will be provided during each class session|
|Class 8||Evaluation of understanding||Details will be provided during each class session|
Elias Stein, Rami Shakarchi "Fourier analysis" Nippon Hyoron sha
Students' course scores are based on final exam 50% and midterm exam 50%.
Students are expected to have passed Calculus I/Recitation and Calculus II/Recitation.