2025 (Current Year) Faculty Courses School of Science Undergraduate major in Mathematics
Applied Analysis I
- Academic unit or major
- Undergraduate major in Mathematics
- Instructor(s)
- Kai Koike
- Class Format
- Lecture
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - Class
- -
- Course Code
- MTH.C211
- Number of credits
- 100
- Course offered
- 2025
- Offered quarter
- 3Q
- Syllabus updated
- Mar 19, 2025
- Language
- Japanese
Syllabus
Course overview and goals
In this course, followed by “Applied Analysis II” (4Q), we learn the basics of Fourier analysis, in particular, the theory of Fourier series. Fourier analysis is a fundamental tool in today’s science and is also an origin of many important ideas in modern mathematics.
One of the primary aims of this course is to learn the basics of the theory of Fourier series. In particular, we learn how to rigorously treat the problem of convergence of Fourier series — when and in what sense do a Fourier series of a function converges? Another aim of the course is to learn applications of Fourier series. In this course, amongst numerous applications (such as to probability theory, analytic number theory, signal and image processing, and so on), we treat applications to differential equations. Another important aim of the course is to allow students to better understand some ideas of modern analysis through the study of Fourier series. For example, the epsilon-delta definition of limits we learned in “Introduction to Analysis I & II” is vital in the understanding of the convergence problem of Fourier series. Moreover, the idea of function spaces, which is an important point of view in the theory of Fourier series, opens the way to the theory of Functional Analysis.
Course description and aims
1) To be able to state the definition of Fourier series and to compute Fourier series of some functions.
2) To be able to prove theorems on convergence of Fourier series and apply them to concrete examples.
3) To be able to apply the technique of Fourier series to solve linear partial differential equations and provide necessary mathematical arguments involved.
Keywords
Fourier series, Heat equation, Dirichlet kernel, Convolution
Competencies
- 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/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Solution method of the heat equation by Fourier series | To be able to use the technique of separation of variables and the principle of superposition to solve the heat equation in terms of trigonometric series (mathematical rigour is not required at this point). |
| Class 2 | Examples of Fourier series | To be able to compute Fourier series of simple functions. |
| Class 3 | Convergence theorems for Fourier series: Smoothness of functions and decay of Fourier coefficients | To be able to explain why twice continuously differentiable functions can be expanded in Fourier series. |
| Class 4 | Convergence theorems for Fourier series: Dirichlet kernel | To be able to derive fundamental properties of the Dirichlet kernel and use them to explain why piecewise continuously differentiable functions can be expanded in Fourier series. |
| Class 5 | Convergence theorems for Fourier series: Fejér kernel | To be able to derive fundamental properties of the Fejér kernel and use them to explain why continuous functions can be uniformly approximated by trigonometric polynomials. |
| Class 6 | Applications of Fourier series (1) | Details will be provided in the class. |
| Class 7 | Applications of Fourier series (2) | Details will be provided in the class. |
Study advice (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)
Evaluation methods and criteria
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.C212 ： Applied Analysis II
- 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
Students are expected to have mastered the contents of the classes "Calculus I / Recitation", "Calculus II", "Calculus Recitation II", "Introduction to Analysis I", and "Introduction to Analysis II".