Master
2021/2022
Fourier analysis and its applications
Type:
Elective course (Systems Analysis and Mathematical Technologies)
Area of studies:
Applied Mathematics and Informatics
Delivered by:
School of Applied Mathematics
When:
1 year, 3 module
Mode of studies:
offline
Open to:
students of all HSE University campuses
Instructors:
Sergey Artamonov
Master’s programme:
Systems Analysis and Mathematical Technologies
Language:
English
ECTS credits:
3
Contact hours:
40
Course Syllabus
Abstract
This program defines the minimal requirements for student competencies, sets up the contents and formats of classes. The program is aimed at lecturers, teaching assistants and students of the specialization 01.04.04 “Applied mathematics” enrolled in the master’s program “Supercomputer simulation in science and engineering”. The program is constructed according to • FGOS VPO on the specialization 01.04.04 “Applied mathematics”. • Working plan of the university on the specialization 01.04.04 “Applied mathematics”. The learning objectives of the development of the discipline "Fourier analysis and its applications" are the formation of students' basic knowledge in the field of modern Fourier analysis and approximation theory, as well as the skills and abilities of their application in various tasks of natural science content; the formation of research skills and the ability to apply them in practice.
Learning Objectives
- The learning objectives of the development of the discipline "Fourier analysis and its applications" are the formation of students' basic knowledge in the field of modern Fourier analysis and approximation theory
Expected Learning Outcomes
- • be able to use mathematical machinery of the course in subsequent learning and professional activities.
- • know basic ideas of Functional analysis, Fourier analysis, theory of functional spaces etc;
- • understand and be able to reproduce proofs of the key course theorems;
Course Contents
- Metric spaces. Linear normed spaces. Eucleadean spaces. Linear operators.
- The space of integrable functions L_1. The Hölder and Minkowski inequalities. L_p spaces
- Approximation by polynomials: Taylor’s theorem, Weierstass’ theorem
- Convolution of functions defined on real line and one dimensional torus.
- Fourier coefficients of periodic functions (definition and properties). Partial sums of Fourie series in terms of the convolution
- Cesaro’s method of summation. Fejer’s theorem The Fourier series and orthogonality. Square integrable functions. The Parseval identity
- Classical Fourier means. General construction of Fourier means.
- The space of Schwartz rapidly decreasing functions on real line and its properties (convolution, pointwise multiplication)
- The Fourier transform of Schwartz rapidly decreasing functions (inversion formula, Parseval identity, Fourier transform and differentiation)
- The Fourier transform of other type of functions (square integrable functions, continuous functions) The distributional approach to the definition of the Fourier transform. Poisson’s summation formula
- Convergence of the Fourier means in Lp spaces.
- Quantitative questions: modulus of continuity and modulus of smoothness
- Theorems of Jackson and Bernstein type
Interim Assessment
- 2021/2022 3rd module0.25 * Аудиторная работа + 0.25 * Домашние задания + 0.5 * Экзамен
Bibliography
Recommended Core Bibliography
- Ряды Фурье; Теория поля; Аналитические и специальные функции; преобразование Лапласа : уч. пособие, Романовский, П. И., 1980
Recommended Additional Bibliography
- Седлецкий, А. М. Классы аналитических преобразований Фурье и экспоненциальные аппроксимации : учебное пособие / А. М. Седлецкий. — Москва : ФИЗМАТЛИТ, 2005. — 504 с. — ISBN 5-9221-0611-2. — Текст : электронный // Лань : электронно-библиотечная система. — URL: https://e.lanbook.com/book/59400 (дата обращения: 00.00.0000). — Режим доступа: для авториз. пользователей.