北京雁栖湖应用数学研究院 北京雁栖湖应用数学研究院

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关于我们
院长致辞
理事会
协作机构
参观来访
人员
管理层
科研人员
博士后
来访学者
行政团队
学术研究
研究团队
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讨论班
招生招聘
教研人员
博士后
学生
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交通
配套设施
周边旅游
新闻
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通知公告
资料下载
清华大学 "求真书院"
清华大学丘成桐数学科学中心
清华三亚国际数学论坛
上海数学与交叉学科研究院
BIMSA > Introduction to wavelets \(ICBS\)
Introduction to wavelets
Wavelet theory is a part of time frequency analysis which allows the study of the local properties of functions by convolving them with shrinking windows and thus describing the structure of functions on various scales, such as fractals for example.
Nowadays, wavelets are widely used in applied sciences (signal and image analysis, data compression, filter banks) as well as in “pure” disciplines (singular operators, functional spaces).
The goal of the course is to give an introduction to the subject in order to provide the students with tools for conducting their own research using the wavelet techniques.
讲师
尤里·柳巴尔斯基
日期
2023年09月19日 至 12月14日
位置
Weekday Time Venue Online ID Password
周二,周四 09:50 - 11:25 A3-2-303 ZOOM 07 559 700 6085 BIMSA
修课要求
Basics of Fourier analysis and Hilbert spaces, however I’ll remind the main notions and facts as soon as needed.
课程大纲
Introduction
- Basic facts from Fourier analysis
- Uncertainty principles
- Ideas of time frequency analysis
- Various types of time-frequency representations

Continuous wavelets transform
- Definition. Reconstruction formula, energy preservation
- Hilbert spaces with reproducing kernel
- Digression: Space of bandlimited signals, Shannon sampling
- Lipschitz regularity
- Local regularity measurements via the Wavelet transform
- Detection of singularities
- Wavelet maxima for images
- Self-similarity. Multifractals.

Discrete wavelets
- Motivation, examples
- Multiresolutional analysis
- Conjugate mirror filters
- Construction of wavelets
- Choosing the filter
- Compactly supported wavelets
- 2D wavelets, image compression
- Wavelets on an interval

Functional spaces
- Survey: Functional spaces, Unconditional basis
- Survey: Tools from real analysis: maximal functions, Marzinkevich interpolation theorem, Calderon- Zygmund operators, Khinchin inequality ( I plan to formulate the main results without proofs, yet if time permitted more details will be given).
- Wavelet basis in L^p spaces
- Wavelet basis in Sobolev spaces
参考资料
We will not follow one textbook. However, the main textbooks are:
1. Mallat, Stéphane, A wavelet tour of signal processing. The sparse way. Elsevier/Academic Press, Amsterdam, 2009.
2. Daubechies, Ingrid, Ten lectures on wavelets. CBMS-NSF Regional Conference Series in Applied Mathematics, 61. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992.
3. Meyer, Yves, Wavelets and operators. Cambridge Studies in Advanced Mathematics, 37. Cambridge University Press, Cambridge, 1992.
听众
Advanced Undergraduate , Graduate
视频公开
公开
笔记公开
公开
语言
英文
讲师介绍
Yurii Lyubarskii received PhD degree in mathematics in 1974 in the Institute for Low Temperature Physics and Engineering, Ukrainian Academy of Sciences and Doctor degree in St. Petersburg branch of Russian mathematical institute in 1990. He taught at the Norwegian University of Science and Technology and also at St. Petersburg State University. Scientific interests of Yu. Lyubarskii include complex and harmonic analysis and applications to the signal analysis.
北京雁栖湖应用数学研究院
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