Jan Mandel/Math7924s10 Theory of distributions

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This readings class deals with Schwartz distributions, also known as generalized functions. The objective is to gain working knowledge for applications. So there will be more problems and discussion and less lecturing.

To discuss online, please go to the talk page.

As always, I will be taking photos of all boards.

Contents

Books

We will read selected chapters from

  • Distribution theory and transform analysis: An introduction, Zemanian, A. H.
  • additional recommended literature:
    • J. Necas: Les methodes directes en theorie des equations elliptiques
    • Generalized functions: theory and applications, Kanwal, R.P.
    • W.F. Donoghue, Distributions and Fourier transforms
    • D.D. Haroske and H. Triebel, Distributions, Sobolev spaces, Elliptic equations

Tentative topics

  • Definition and basic properties (ch. 1)
  • Convergence of distributions (ch. 2)
  • Distributions as derivatives of continuous functions (ch. 3)
  • Fourier transform and tempered distributions (a.k.a. of slow growth) (ch. 7)
  • Distributions and test functions as locally convex topological spaces (from Donoghue)
  • Sobolev spaces and applications to partial differential equations (from Necas and Haroske/Triebel)
  • Distribution derivatives of functions with jump discontinuities (from Kanwal)

Prerequisites

  • Real analysis (Lebesgue integration,...), Functional analysis (Banach spaces,...)

Requirements

To receive credit students must give a presentation and turn in a written preparation.

Registration

Use hand registration form, course Math 7924, 1 credit hour, 5 hours of work per week, copy the information above including the Requirements.

Notes

  • Aug 30: Introduction, definition of distributions (1.1-1.3)
  • Sep 7: Labor Day
  • Sep 13: Basic properties (1.6-1.8, 2.1-2.2 convergence of distributions))
  • Sep 20: Examples of convergence of distributions (2.3), Hw: 2.2/4, 2.3/4a,9
  • Sep 27: Differentiation of distributions (2.4), Indefinite integral of distribution (2.6). Hw: 2.4/1,14
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