Generalized Functions

Below are video files that were recorded with the support of Shanghai Jiao Tong University (SJTU) in 2014. The files sizes can be quite large. The download links will open in a new window, asking you to confirm download from the SJTU servers.

The videos are copyrighted; in particular, you do not have permission to upload them to any other video sharing site (such as YouTube, Vimeo, etc.). They are intended for personal use and viewing only.

The usual caveat applies: at the time of recording, inevitably, minor typographical errors were also incorporated. These have not been corrected within the video files. However, the PDF slides corresponding to each lecture can be downloaded further below. There, known errors have been corrected.  Further corrections may be sent to horst@sjtu.edu.cn.

1. Introduction

What are point sources? A first attempt at a mathematical description. How can point source approximations be used for general solutions of differential equations? A first example of a Green function.

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2. Generalized Functions (Distributions)

Lots of basic concepts: Functions of compact support. Null sequences and continuous linear maps. Locally integrable functions. The delta "function". Regular and singular distributions.

  • Explain what a test function is.
  • Explain what a null sequence of test functions is and give an example and a counter-example.
  • Explain what a distribution is.
  • What is a locally integrable function? Give examples and counter-examples.
  • Define what a regular and a singular distribution is. Give examples.
  • Download Assignment 1 and try to solve the exercises.
  • Do some of the Exercises in Stakgold /Holst for this section.








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3. Operations on Distributions

Working with distributions: basic operations such as translation (shifting), dilating (scaling). Can distributions be multiplied? Differentiating distributions.  The logarithm and the principle value of 1/x. The Laplacian of 1/|x| in three-dimensional space.

  • Explain how operations for functions are extended to distributions ``by duality''.
  • Explain how to differentiate a non-differentiable function.
  • Why is the real function g given by g(x)=1/x not a distribution? Explain what the Cauchy principal value of g is as an ordinary function (if necessary, find this out from an internet search) and how it is interpreted as a "principal value distribution."
  • Download Assignment 2 and try to solve the exercises.
  • Do some of the Exercises in Stakgold /Holst for this section.







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4. Families of Distributions

Sequences and general one-parameter families of distributions. Null sequences of distributions. Convergence and delta families.

  • How is the convergence of a family of distributions defined?
  • What is a delta sequence or a delta family? Give examples!
  • Download Assignment 3 and try to solve the exercises.
  • Do some of the Exercises in Stakgold /Holst for this section.







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5. The Classical Fourier Transform

Functions of rapid decrease (Schwartz functions). The classical Fourier transform of Schwartz functions and its continuity. The inversion theorem for the Fourier transform on Schwartz functions. Further properties of the Fourier transform. Convolution of Schwartz functions.

  • How is the space of Schwartz functions defined?
  • What is the relationship between the sets of test functions and Schwartz functions?
  • Give three examples (not trivially equivalent) of Schwartz functions.
  • How is the Fourier transform defined for Schwartz functions?
  • How is convergence defined in the space of Schwartz functions? What does "continuity of the Fourier transform" mean?
  • List the basic properties of the Fouriertransform and the convolution.
  • Download Assignment 4 and try to solve the exercises.
  • Do some of the Exercises in Stakgold /Holst for this section.




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6. Tempered Distributions and the Fourier Transform

Tempered distributions as a special case of distributions. The Fourier transform for tempered distributions with examples. A major application: solving the homogeneous heat equation in n+1 dimensions with an initial condition as a tempered distribution.

  • What is a tempered distribution?
  • How is the Fourier transform defined for tempered distributions?
  • Explain how the convolution can be defined for tempered distributions.
  • Download Assignment 5 and try to solve the exercises.
  • Do some of the Exercises in Stakgold /Holst for this section.