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- Academic unit or major
- Undergraduate major in Mathematics
- Instructor(s)
- Koike Kai
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- Day/Period(Room No.)
- Wed3-4(H112)
- Group
- -
- Course number
- MTH.C212
- Credits
- 1
- Academic year
- 2022
- Offered quarter
- 4Q
- Syllabus updated
- 2022/4/20
- 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”. In the first part of the course, we revisit the theory of Fourier series from a new point of view, that is, from the perspective of functional analysis. More concretely, we shall learn how Fourier series expansion can be viewed as expansion by an orthonormal basis of a “space of square-integrable functions”. In the second part, we learn the theory of Fourier transforms which is a variation of Fourier series for functions defined on the whole real line. This opens the way to a broader range of applications of the idea of Fourier analysis.
One of the primary aims of the course is to introduce the students some basic notions of functional analysis by revisiting the theory of Fourier series from functional analytic point of view (more thorough treatment will be given in “Functional Analysis” and “Real Analysis I & II”). Another objective of the course is to learn the basics of the theory of Fourier transforms together with its applications to differential equations.
Student learning outcomes
1) To be able to explain the definitions of inner product and Hilbert spaces and their elementary properties together with suitable examples (such as spaces of square-integrable functions).
2) Understand a functional analytic view of Fourier series and to be able to derive Bessel's inequality. Also, understand Parseval's equality and Riemann−Lebesgue lemma and to be able to use them to evaluate certain series and integrals.
3) Understand basic properties of Fourier transforms and to be able to compute several examples. Also, to understand inverse Fourier transforms and the Fourier inversion formula and to be able to apply them, for example, to differential equations.
Keywords
Spaces of square-integrable functions, inner product and Hilbert spaces, orthonormal basis, Bessel's inequality, Parseval's equality, Riemann−Lebesgue lemma, Fourier transforms and inverse Fourier transforms, Fourier inversion formula
Competencies that will be developed
| ✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Class flow
The course will be based on class handouts and lecture notes. Assignments (problem sets) to enhance understanding of the lecture are given.
Course schedule/Required learning
| Course schedule | Required learning | |
|---|---|---|
| Class 1 | Spaces of square-integrable functions; inner product and Hilbert spaces | Understand the definition of spaces of square-integrable functions and to be able to provide (counter)examples. Also to be able to prove elementary properties of inner product and Hilbert spaces. |
| Class 2 | Mean-square convergence of Fourier series (1) | To be able to prove Bessel's inequality based on elementary properties of inner product spaces. Also, understand what Parseval's equality and Riemann−Lebesgue lemma are and to be able to apply them to evaluation of certain series and integrals. |
| Class 3 | Mean-square convergence of Fourier series (2) | Same as Class 2. |
| Class 4 | Fourier transform and its fundamental properties | To be able to state the definition of Fourier transforms and prove elementary properties (such as their relation to differentiation). |
| Class 5 | Examples of Fourier transform | To be able to compute some examples of Fourier transforms. |
| Class 6 | Fourier inversion formula | To be able to write down the definition of the inverse Fourier transforms and the Fourier inversion formula. |
| 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 each for preparation and review referring to this syllabus and other course materials (more detailed instructions shall be given when needed).
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 testing the understanding of the contents of each class and a term-end exam assessing the achievement of the course aim. Some advanced knowledge beyond those described in the aim of the course may also be evaluated (details are given during the course).
Assignments (30%), Term-end exam (70%)
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".
