Type of Credit: Elective
Credit(s)
Number of Students
本課程會介紹傅里葉級數以及傅里葉變換的數學理論，傅里葉級數可視作離散版本的傅里葉變換。藉由傅里葉分析，本課程也會介紹“廣義函數”，例如 Dirac delta “函數”（儘管名爲函數，事實上它不是一個真正的函數）。本課程是現代分析的基礎課程。
We will introduce the mathematical theory of Fourier series and Fourier transforms. The Fourier series can be viewed as a discrete form of the Fourier transform. Using Fourier analysis, we will also explore distributions (generalized functions), such as the Dirac delta 'function', which, despite its name, is not a true function. This course serves as an introductory foundation in modern analysis.
能力項目說明
本課程會用現代分析來傳授古典的傅里葉分析，讓學生瞭解古典分析與現代分析的關係。
We will use modern analysis to teach classical Fourier analysis, helping students understand the connection between classical and modern analysis.
本大綱只供參考，課程進行中會隨時更新。
This syllabus is just for a reference, which will be updated at any time during the course.
期中考試（預計第7週）及期末考試日期（預計第16週）視教學進度而定。
The date of midterm (expect the 7th week) and final exams (expect the 16th week) will depend on the actual situation.
我們預計根據下列計劃進行教學：
We will plan to deliver the ideas based on the followings:
| 課程主題 | 課程内容與指定閲讀 | 教學活動與作業 | |
|
第一週 1st week |
Fourier series | Weak derivatives |
教師授課 佈置作業 |
|
第二週 2nd week |
Fourier series | Fourier series in L2 |
教師授課 佈置作業 |
|
第三週 3rd week |
Fourier series | Fourier series in L2 |
教師授課 佈置作業 |
|
第四週 4th week |
Fourier series |
Pointwise and uniform convergence Dini's criterion |
教師授課 佈置作業 |
|
第五週 5th week |
Fourier series |
Gibbs-Wilbraham phenomenon Cesàro summability of Fourier series |
教師授課 |
|
第六週 6th week |
Fourier series | 復習 | 教師授課 |
|
第七週 7th week |
期中考試 Midterm exam | ||
|
第八週 8th week |
Fourier transform | Fourier transform on Schwartz space |
教師授課 佈置作業 |
|
第九週 9th week |
Fourier transform | Fourier transform on Tempered distributions |
教師授課 佈置作業 |
|
第十週 10th week |
Fourier transform | Fourier transform on L2 |
教師授課 佈置作業 |
|
第十一週 11th week |
Fourier transform |
Compactly supported distributions test function and distributions |
教師授課 佈置作業 |
|
第十二週 12th week |
Fourier transform |
Convolution of functions and distributions Convolution between distributions |
教師授課 佈置作業 |
|
第十三週 13th week |
Fourier transform |
Convolution of distributions Structure of distribution |
教師授課 佈置作業 |
|
第十四週 14th week |
Fourier transform |
Fourier transform on compactly supported Convolution theorem Convolution of pv tempered distributions |
教師授課 |
|
第十五週 15th week |
Fourier transform |
復習 |
教師授課 |
|
第十六週 16th week |
期末考試 Final exam |
每週會有3節正課。建議每週花至少3小時自習。
There will be 3 regular classes per week. It is recommend to spend at least 3 hours for self-study.
作業 Homework 60%
期中考試 Midterm exam 20%
期末考試 Final exam 20%
*考試形式將視修課情形再決定
自備講義：
https://puzhaokow1993.github.io/homepage/teaching/Lecture_Note/ver1_Lecture_Note_Distributions_Fourier_Analysis.pdf
https://puzhaokow1993.github.io/homepage/teaching/courses_pages/Fourier_Analysis_751799001_701866001_2025_Fall/Fourier_Analysis_751799001_701866001_2025_Fall.html