2018 | Applied Functional Analysis

2018　Applied Functional Analysis

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Academic unit or major: Graduate major in Mathematical and Computing Science

Instructor(s): Miura Hideyuki Nishibata Shinya Umehara Masaaki Terashima Yuji Murofushi Toshiaki Suzuki Sakie

Class Format: Lecture

Media-enhanced courses

Day/Period(Room No.): Mon7-8(W832) Thr7-8(W832)

Group: -

Course number: MCS.T409

Credits: 2

Academic year: 2018

Offered quarter: 2Q

Syllabus updated: 2018/3/20

Lecture notes updated: -

Language used: Japanese

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Syllabus

Course description and aims

The fundamentals of the functional analysis are given for the application to the mathematical and computing sciences. Students understand the theory of function spaces such as Lebesgue spaces and Sobolev spaces and the Fourier transform. They are able to apply them to the partial differential equations.

Student learning outcomes

This course emphasizes the importance of rigorous treatment of various problems in mathematical and computing sciences by the use of concepts in the functional analysis. In particular students are able to understand the fundamentals of the operator theory, the Fourier transform and the theory of distributions, and apply them to partial differential equations and so on.

Keywords

Function spaces, Inequalities for functions, Fourier transform, Distributions, Partial differential equations

Competencies that will be developed

✔ Specialist skills

Intercultural skills

Communication skills

Critical thinking skills

✔ Practical and/or problem-solving skills

Class flow

After preliminaries of topology, the lecture is devoted to the fundamentals to the functional analysis. The Fourier transform and the theory of distributions is given for the applications to the partial differential equations. In order to cultivate a better understanding, some exercises are given.

Course schedule/Required learning

	Course schedule	Required learning
Class 1	Banach spaces and examples	Understand the contents in the lecture.
Class 2	Lebesgue spaces	Understand the contents in the lecture.
Class 3	Inequalities in function spaces	Understand the contents in the lecture.
Class 4	Convolution and mollifiers	Understand the contents in the lecture.
Class 5	Properties of the Fourier transform	Understand the contents in the lecture.
Class 6	Fourier inversion formula	Understand the contents in the lecture.
Class 7	Properties of rapidly decaying functions	Understand the contents in the lecture.
Class 8	Distributions	Understand the contents in the lecture.
Class 9	Tempered distributions and the Fourier transform	Understand the contents in the lecture.
Class 10	Derivatives of distributions and the Sobolev spaces	Understand the contents in the lecture.
Class 11	Sobolev's embedding theorem and Rellich's compactness theorem	Understand the contents in the lecture.
Class 12	Convolution of distributions	Understand the contents in the lecture.
Class 13	Applications to the Laplace equation	Understand the contents in the lecture.
Class 14	Application to the heat equation	Understand the contents in the lecture.
Class 15	Applications to the wave equation	Understand the contents in the lecture.

Textbook(s)

To be specified in the first lecture.

Reference books, course materials, etc.

Unspecified.

Assessment criteria and methods

Learning achievement is evaluated by reports.

Related courses

ZUA.B201 ： Set and Topology I
ZUA.B203 ： Set and Topology II

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

Students should understand the fundamentals of topology and measure theory.

TOKYO INSTITUTE OF TECHNOLOGY