2020/2021
Гармонический анализ и теория приближений
Лучший по критерию «Полезность курса для Вашей будущей карьеры»
Лучший по критерию «Полезность курса для расширения кругозора и разностороннего развития»
Статус:
Дисциплина общефакультетского пула
Кто читает:
Факультет математики
Где читается:
Факультет математики
Когда читается:
3, 4 модуль
Преподаватели:
Тихонов Сергей Юрьевич
Язык:
английский
Кредиты:
6
Контактные часы:
72
Course Syllabus
Abstract
The course is designed for undergraduate, master's, and postgraduate students. Knowledge requirements - standard courses in mathematical analysis and, preferably (but not necessarily), in functional analysis. The course materials can be divided into two groups: the first - basic concepts and results in Fourier analysis, the second - modern approaches in approximation theory and harmonic analysis
Learning Objectives
- Teach the student to work with the basic concepts of harmonic analysis and give, if possible, a full range of problems and methods for solving them in approximation theory and Fourier analysis.
Expected Learning Outcomes
- Hardy-Littlewood Fractional Integral Theorems, Sobolev Embedding Theorems, Uncertainty Principle for Fourier Transforms.
Course Contents
- Basic concepts and problems of approximation theory.
- Polynomial inequalities and discretization.
- Measures of smoothness of functions.
- Fourier series and transformations.
- Function spaces (Lebesgue, Lorentz, Sobolev, and Besov), embedding theorems
- Operator interpolation.
- The uncertainty principle.
- (*) Compressed sensing.
Interim Assessment
- Interim assessment (4 module)0.5 * final exam + 0.3 * Presentations/reports + 0.2 * tests
Bibliography
Recommended Core Bibliography
- Carl De Boor, Ronald A. Devore, & Amos Ron. (n.d.). CONSTRUCTIVE APPROXIMATION 9 1993 Springer-Verlag New York Inc. On the Construction of Multivariate (Pre)Wavelets. Http://Www.Math.Tamu.Edu/~rdevore/Publications/75.Pdf.
Recommended Additional Bibliography
- Loukas Grafakos. (2014). Modern Fourier Analysis (Vol. 3rd ed. 2014). Springer.