2020 Faculty Courses School of Science Undergraduate major in Mathematics
Applied Analysis II
- 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.C212
- Number of credits
- 100
- Course offered
- 2020
- Offered quarter
- 4Q
- Syllabus updated
- Jul 10, 2025
- Language
- Japanese
Syllabus
Course overview and goals
The first half of this course is devoted to study the functional analytic framework of Fourier series.
The second half is devoted to study Fourier transform, which is a continuous counterpart of Fourier series.
In particular, we reconsider Fourier series as an orthonormal basis in a function space, and understand an abstract framework of Fourier series.
Moreover, we study fundamentals of Fourier transform and its applications to differential equations.
Course description and aims
Students are expected to understand the functional analytic framework of Fourier series.
Moreover, we aim at understanding fundamentals of Fourier transform, its relation to Fourier series, and applications to differential equations.
Keywords
Hilbert space, orthonormal basis, Bessel's inequality, Parseval's identity, Fourier transform, Riemann-Lebesgue lemma, Fourier inversion formula
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 | Function spaces | Details will be provided during each class session |
| Class 2 | Examples of function spaces | Details will be provided during each class session |
| Class 3 | Fourier series and orthonormal basis | Details will be provided during each class session |
| Class 4 | Fourier transform and its fundamental properties | Details will be provided during each class session |
| Class 5 | Examples of Fourier transform | Details will be provided during each class session |
| Class 6 | Fourier inversion formula | Details will be provided during each class session |
| Class 7 | Applications of Fourier transform | 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 homework (30%)
Related courses
- ZUA.C201 ： Advanced Calculus I
- ZUA.C203 ： Advanced Calculus II
- MTH.C211 ： Applied Analysis I
- MTH.C301 ： Complex Analysis I
- MTH.C302 ： Complex Analysis II
Prerequisites
Students are expected to have passed Calculus I/Recitation and Calculus II/Recitation.