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Regular version of the site
Master 2019/2020

## Fourier analysis and its applications

Area of studies: Applied Mathematics
When: 2 year, 1, 2 module
Mode of studies: offline
Instructors: Sergey Artamonov
Master’s programme: Supercomputer Simulations in Science and Engineering
Language: English
ECTS credits: 6

### 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

• • 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;
• • be able to use mathematical machinery of the course in subsequent learning and professional activities. #### Course Contents

• The space of integrable functions L_1. The Hölder and Minkowski inequalities. L_p spaces
• Approximation by polynomials: Taylor’s theorem, Weierstass’ theorem
• Approximation in norm linear spaces: convexity and the best approximation problem
• 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
• 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 integrable functions (Riemann’s lemma, Fourier transform of the convolutions)
• 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
• Quantitative questions: modulus of continuity and modulus of smoothness
• Theorems of Jackson and Bernstein type #### Assessment Elements

• самостоятельные работы
• домашние задания
• экзамен
• Контрольно-измерительные материалы #### Interim Assessment

• Interim assessment (2 module)
0.25 * домашние задания + 0.25 * самостоятельные работы + 0.5 * экзамен #### Recommended Core Bibliography

• Ряды Фурье; Теория поля; Аналитические и специальные функции; преобразование Лапласа : уч. пособие, Романовский П. И., 1980