2019 Applied Analysis II

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Academic unit or major
Undergraduate major in Mathematics
Instructor(s)
Kawahira Tomoki 
Course component(s)
Lecture
Day/Period(Room No.)
Wed3-4(H112)  
Group
-
Course number
MTH.C212
Credits
1
Academic year
2019
Offered quarter
4Q
Syllabus updated
2019/3/18
Lecture notes updated
-
Language used
Japanese
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Course description and aims

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.

Student learning outcomes

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 that will be developed

Intercultural skills Communication skills Specialist 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/Required learning

  Course schedule Required learning
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
Class 8 Evaluation of understanding Details will be provided during each class session

Textbook(s)

None required

Reference books, course materials, etc.

Elias Stein, Rami Shakarchi "Fourier analysis" Nippon Hyoron sha

Assessment criteria and methods

In-class exercises (25 %) and homework (75 %)

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 (i.e., required knowledge, skills, courses, etc.)

Students are expected to have passed Calculus I/Recitation and Calculus II/Recitation.

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