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2022　Applied Analysis I

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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.C211
Credits
1
Academic year
2022
Offered quarter
3Q
Syllabus updated
2022/4/20
Lecture notes updated
-
Language used
Japanese
Access Index

Course description and aims

　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 some 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 some 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 learn in “Introduction to Analysis I & II” is vital in the understanding of the convergence problem of Fourier series. Moreover, the idea of "function spaces" treated in "Applied Analysis II", which is also essential in the theory of Fourier series, opens the way to the theory we learn in “Functional Analysis”.

Student learning outcomes

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 Dirichlet's theorem on convergence of Fourier series and apply it to some examples. Also to be able to give important examples concerning (non)convergence of Fourier series.
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, series of functions, pointwise and uniform convergence, Gibbs phenomenon, Dirichlet kernel, the Dirichlet conditions, heat and wave equation

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 Solution methods of partial differential equations and trigonometric series expansion of functions To be able to use the technique of separation of variables and the principle of superposition to solve linear partial differential equations in terms of trigonometric series (mathematical rigour is not required at this point).
Class 2 Complex-valued functions; convergence of series of functions To be able to calculate basic operations on complex-valued functions (in particular complex exponentials). To be able to state the epsilon-delta definitions of pointwise and uniform convergence of series of functions and to be able to give related examples.
Class 3 Examples of Fourier series To be able to compute Fourier coefficients of some elementary functions.
Class 4 Convergence theorems for Fourier series (1) To be able to prove Dirichlet's theorem on convergence of Fourier series and apply it to some examples. Also to be able to give important examples concerning (non)convergence of Fourier series.
Class 5 Convergence theorems for Fourier series (2) Details will be provided in the class.
Class 6 Applications of Fourier series (1) To be able to use the solution method of linear partial differential equations explained in Class 1 with mathematical rigour.
Class 7 Applications of Fourier series (2) 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.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 (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", and "Introduction to Analysis II".

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