• Generalized Functions
    • Department of Mathematics
    • Credit. 2
    • MA421
    • Enroll
    • Fall , 2015
    • 3946
    • Course Description:
    • ( Exchange Programme )
    • The course is an introduction to the theory of generalized functions (distributions) as functionals defined on certain classes of test functions. Starting with examples such as the Dirac delta-function I will talk about the basic properties of generalized functions including local properties and regularization of divergent integrals. Then I will also talk about differentiation and integration of generalized functions, and their application to divergent Fourier series, and to solving ODE and PDE. Then we will consider the convolution of generalized functions and Fourier and Laplace transform. We will also consider Abel’s integral equation and fractional derivatives.
      After completing the course, students should know the definition and basic (analytic) properties of generalized functions and the spaces of generalized functions, and be able to apply them in different areas of mathematics such as ODE, PDE, and integral equations.
    • Course Syllabus:
    • Syllabus (tentative and subject to change if any):
      1. Definition of generalized functions and their basic properties. The space of generalized functions.
      2. Diraс delta-function and delta-sequences
      3. Regular generalized function, Sokhotski formulas
      4. Differentiation of generalized functions
      5. Direct product of generalized functions
      6. Convolution of generalized functions
      7. Integrals and derivatives of high orders
      8. The Poisson summation formula
      9. Fourier Transform of generalized functions.
      10. Laplace transform of generalized
      11. Some elements of the Sobolev spaces theory
    • Schedule:
  • Reading list
  • Other Materials
  • Discussion
  • Homework download/submit
    • Mikhail Tyaglov
    • Read more
    • Male
    • E-mail:
    • tyaglov@sjtu.edu.cn
    • Profile
  • Prerequisite Course:

  • Textbooks:

    1. Vladimirov V., Equations of mathematical physics, Marcel Dekker, inc., New York, 1971
    2. Kanwal R., Generalized functions. Theory and applications, 3d ed., Springer Science + Business Media, LLC, 2004
    3. Gelfand I. and Shilov G., Generalized functions, vol. I, Academic press, New York and London, 1964
  • Grading:

    The grading criteria: the students will have 12–16 homeworks (depending on the progress of students during the class). Completely made homeworks give 60% of the final mark, and the final work gives other 40%.
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