2026Year

First Semester

Applied Analysis

Lecture

Applied Analysis

Z1500010 Applied Analysis

HASEGAWA Kenji

2.0Credits

Tue.1Period

Shinjuku Remote

A) A high degree of specialized expertise　100%
B) The skills to use science and technology　0%
C) The ability to conduct research independently, knowledge pertaining to society and occupations, and sense of ethics required of engineers and researchers　0%
D) Creative skills in specific areas of specialization　0%

Fourier analysis is a powerful mathematical method centered on Fourier series, which represent periodic functions as linear combinations of trigonometric functions, and the Fourier transform, which decomposes functions into their frequency components. These methods have had a significant impact on engineering and information science.

In this course, we study the theoretical foundations of Fourier analysis. In addition, we introduce orthonormal function systems based on classical polynomial families. Furthermore, as a method that complements the difficulty of capturing local properties of functions by Fourier series and Fourier transforms, we also discuss the relatively recent theory of wavelet transforms.

Upon successful completion of this course, students will be able to:

(1) Understand the mathematical properties of Fourier series and Fourier transforms and explain their application to differential equations and related problems.
(2) Understand the general theory of orthonormal function systems and explain classical examples.
(3) Understand the basic principles of wavelet transforms.

Understanding of calculus, integral calculus, and complex functions taught at the undergraduate level is necessary.
It is desirable to be able to use computer algebra systems (such as Matlab, Mathematica, Maxima, etc.) to solve short test and report problems.

Support for self-learning using ICT

1. Guidance
2. Fourier Integral and Fourier Series
3. Dirichlet Kernel and Convergence of Fourier Series
4. L^2 Norm and Inner Product
5. Mean Convergence
6. Fourier Transform
7. Inversion Formula
8. Delta Function and Heaviside Function
9. Orthogonal Function Systems
10. Legendre Polynomials, Hermite Polynomials, Laguerre Polynomials
11. Continuous Wavelet Transform
12. Discrete Wavelet Transform
13. Scaling Function
14. Meyer Wavelet
15. Daubechies Wavelet
Students will watch the slides and submit their answers to the short tests by the deadline. A report assignment will be given at the end of the semester.

The final grade for this course will be determined based on short tests and a report administered through KU-LMS, with a weighting of Short Tests : Report = 2 : 1. A short test or the report will be deemed invalid in the following cases:

(1) Failure to complete attendance registration and download the course materials by the designated deadline.
(2) When a student is judged to have copied, or allowed others to copy, answers from another student or from generative AI tools.
(3) Failure to download the report assignment file by the designated deadline.
(4) When materials other than the course materials are consulted without properly citing the source and submitting an image of the relevant page.

Evaluation of short tests and the report will be based not only on correctness but also on learning records in KU-LMS (including attendance registration, access/download history of materials, submission time, etc.). If the report is not submitted, or is deemed invalid, the final grade for the course will be recorded as F.

The correct answers for the short test will be disclosed through KU-LMS. In addition to office hours, the teacher will respond to questions and opinions on Google Meet. The date and time will be discussed via email (ft10058@g.kogakuin.jp).

There is no specified textbook. Instead, students will use the PDF files uploaded to KU-LMS for study materials.

Treatises on Fourier analysis, applied analysis, and wavelets.

Wednesday, 13:00〜14:00(Hachioji:1E-313)

It is an on-demand remote class, chosen as a teaching method deemed more effective than face-to-face interaction. Unless there is a change in instructors, the class format will be maintained going forward.

Not applicable

Electrical Engineering and Electronics Program／Informatics Program