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- Academic unit or major
- Undergraduate major in Mathematics
- Instructor(s)
- Koike Kai
- Class Format
- Lecture
- Media-enhanced courses
- Day/Period(Room No.)
- -
- Group
- -
- Course number
- MTH.C212
- Credits
- 1
- Academic year
- 2024
- Offered quarter
- 4Q
- Syllabus updated
- 2024/3/14
- Lecture notes updated
- -
- Language used
- Japanese
- Access Index
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Course description and aims
This course is a continuation of “Advanced Calculus I”. We first revisit the theory of Fourier series from a geometric view point. More concretely, we shall learn how Fourier series expansion can be viewed as expansion by an "orthonormal basis". Next, we learn the theory of Fourier transforms on the real line which is a variation of Fourier series for functions defined on the real line. This opens the way to a broader range of applications of the idea of Fourier analysis.
The aim of the course is to learn the basics of the theory of Fourier transforms on the real line together with its applications to differential equations.
Student learning outcomes
1) To be able to explain geometric ideas behind Fourier series (inner products, orthonormal basis); to be able to understand and use Parseval's theorem.
2) Understand basic properties of Fourier transforms on the real line and to be able to compute several examples.
3) To be able to use the Fourier inversion formula and Plancherel's theorem.
Keywords
Orthonormal basis, Parseval's theorem, Fourier transform on the real line, Fourier inversion formula, Plancherel's theorem
Competencies that will be developed
| ✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Class flow
Assignments (problem sets) to enhance understanding of the lecture are given.
Course schedule/Required learning
| Course schedule | Required learning | |
|---|---|---|
| Class 1 | Orthonormal basis and Fourier series | Understand the notion of inner products of functions; to be able to explain how Fourier series expansion can be viewed as expansion in terms of orthonormal basis. |
| Class 2 | Mean-square convergence of Fourier series and Parseval's theorem | To be able to explain the mean-square convergence of Fourier series with the geometric view point explained earlier; to be able to explain what Parseval's theorem is and learn some of its applications. |
| Class 3 | Fourier transform on the real line and its fundamental properties | To be able to state the definition of Fourier transforms on the real line and prove elementary properties (such as their relation to differentiation). |
| Class 4 | Fourier inversion formula | To understand the Fourier inversion formula and some of its applications. |
| Class 5 | Fourier transform and convolution | To be able to explain the relation between Fourier transforms and convolutions. |
| Class 6 | Plancherel's theorem | To understand Plancherel's theorem and some of its applications. |
| Class 7 | Applications of Fourier transform | Details will be provided in the class. |
Out-of-Class Study Time (Preparation and Review)
Before and after each class, students should spend approximately 100 minutes or more for preparation.
Textbook(s)
None required
Reference books, course materials, etc.
Elias M. Stein and Rami Shakarchi, Fourier Analysis: An Introduction, Princeton University Press (2003)
Assessment criteria and methods
Evaluation is based on assignments and a term-end exam assessing the achievement of the course aim.
Related courses
- MTH.C201 ： Introduction to Analysis I
- MTH.C202 ： Introduction to Analysis II
- MTH.C211 ： Applied Analysis I
- MTH.C351 ： Functional Analysis
- MTH.C305 ： Real Analysis I
- MTH.C306 ： Real Analysis II
- MTH.C301 ： Complex Analysis I
- MTH.C302 ： Complex Analysis II
Prerequisites (i.e., required knowledge, skills, courses, etc.)
Students are expected to have mastered the contents of the classes "Calculus I / Recitation", "Calculus II", "Calculus Recitation II", "Introduction to Analysis I", "Introduction to Analysis II", and "Applied Analysis I".
