Introductiontoharmonicanalysis

Mukhamedova Raushan Orazgalievna

The instructor profile

Description: A section of mathematical analysis in which the properties of functions are studied by representing them as series or Fourier integrals. Also, a method for solving problems using the representation of functions in the form of series or Fourier integrals. The main objects of study of classical harmonic analysis are trigonometric series, Fourier transform, almost periodic functions, Dirichlet series. Solving problems in Matlab

Amount of credits: 6

Course Workload:

Types of classes hours
Lectures 30
Practical works 30
Laboratory works
SAWTG (Student Autonomous Work under Teacher Guidance) 30
SAW (Student autonomous work) 90
Form of final control Exam
Final assessment method

Component: University component

Cycle: Base disciplines

Goal
  • Increasing the level of professional competence of students, the formation of the concept of the technical capabilities of one of the sections of modern analysis and the role of the Fourier transform in the problems of information security theory.
Objective
  • To study the main technical means of modern harmonic analysis: maximum functions, convolutions, interpolation theorems
  • study the basic properties of the Fourier transform of functions;
  • to prepare students for the use of the Fourier transform in the problems of information security theory.
Learning outcome: knowledge and understanding
  • Properties of maximum functions and approximative units for estimating harmonic analysis operators;
  • Basic properties of the Fourier transform;
  • Theoretical and practical skills of the fundamentals of harmonic analysis in mathematics;
Learning outcome: applying knowledge and understanding
  • Methods for proving the properties of the Fourier transform;
  • Self-education and ways to use the Fourier transform apparatus for mathematical research.
Learning outcome: formation of judgments
  • Integrate knowledge, deal with complexity and make judgments based on incomplete or limited information, taking into account the ethical and social responsibility for the application of these judgments and knowledge;
  • Анализировать и принимать решения по социальным, этическим, научным и техническим проблемам, возникающим в профессиональной деятельности.
  • An idea of ​​approaches to solving non-standard problems and the search for new original ideas and design methods;
Learning outcome: communicative abilities
  • Show initiative and creativity, including in unusual situations.
Learning outcome: learning skills or learning abilities
  • Ability for independent research activities (analysis, comparison, systematization, abstraction, modeling, data validation, decision making, etc.),
  • Ability for continuous self-education;
Teaching methods

The preparation and defense of reports and literature reviews by students on a given topic allows students to expand their scientific horizons, improve their skills in working with educational and scientific domestic and foreign literature, develop language skills, improve mathematical skills, strengthen interdisciplinary ties, develop the skill to systematize and freely present material to an audience on a given topic, to lay the foundation for further research work.

Topics of lectures
  • Гармонический анализ
  • Ортогональные и ортонормированные системы
  • Ортогональный ряд по ортонормированной системе
  • Общий ряд Фурье
  • Тригонометрическая система и ее свойства
  • Признаки сходимости ряда Фурье
  • Теорема о дифференцировании ряда Фурье
  • Тригонометрический интеграл и интеграл Фурье
  • Теорема о непрерывности специального несобственного интеграла, зависящего от параметра
  • Теорема о дифференцировании по параметру специального несобственного интеграла
  • Теорема об интегрировании в конечных пределах по параметру специального несобственного интеграла
  • разложение в ряд Фурье
  • Гармонический анализ и синтез сигналов
Key reading
  • 1.Зорич анализ. Т.2. – М.: Наука, 2005.
  • 2. Садовничий и упражнения по математическому анализу. Книга 2. – М.: Высшая школа, 2010.
Further reading
  • 1.Власова Е.А. Ряды. М. : МГТУ им. Н. Э. Баумана, 2000. – 612 с.
  • 2. Пискунов Н.С. Дифференциальное и интегральное исчисления. Том 2. М.: Наука, 2010.
  • 3. Письменный Д.Т. Конспект лекций по высшей математике. Полный курс. М.: Айрис-пресс, 2011. – 608 с.
  • 4. http://www.mathprofi.ru/
  • 5. http://ru.wikipedia.org
  • 6. http://www.math24.ru/definition-of-fourier-series.html