2020 Faculty Courses School of Science Undergraduate major in Mathematics
Applied Analysis I
- Academic unit or major
- Undergraduate major in Mathematics
- Instructor(s)
- Yoshihiro Tonegawa
- Class Format
- Lecture (Zoom)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 3-4 Wed (H112)
- Class
- -
- Course Code
- MTH.C211
- Number of credits
- 100
- Course offered
- 2020
- Offered quarter
- 3Q
- Syllabus updated
- Jul 10, 2025
- Language
- Japanese
Syllabus
Course overview and goals
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.
Course description and aims
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.
Keywords
Series of functions, Fourier series, Bessel's inequality, Riemann-Lebesgue lemma, Dirichlet kernel
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
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/Objectives
| Course schedule | Objectives | |
|---|---|---|
| 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 |
Study advice (preparation and review)
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
Textbook(s)
None required
Reference books, course materials, etc.
Elias Stein, Rami Shakarchi "Fourier analysis" Nippon Hyoron sha
Evaluation methods and criteria
Final exam (70 %) and homeworks (30 %)
Related courses
- ZUA.C201 ： Advanced Calculus I
- ZUA.C203 ： Advanced Calculus II
- MTH.C212 ： Applied Analysis II
- MTH.C301 ： Complex Analysis I
- MTH.C302 ： Complex Analysis II
Prerequisites
Students are expected to have passed Calculus I/Recitation and Calculus II/Recitation.