Mathematical Preliminaries

  • Kristian Bredies
  • Dirk Lorenz
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


Mathematical image processing, as a branch of applied mathematics, is not a self-contained theory of its own, but rather builds on a variety of different fields, such as Fourier analysis, the theory of partial differential equations, and inverse problems. In this chapter, we deal with some of those fundamentals that commonly are beyond the scope of introductory lectures on analysis and linear algebra. In particular, we introduce several notions of functional analysis and briefly touch upon measure theory in order to study classes of Lebesgue spaces. Furthermore, we give an introduction to the theory of weak derivatives as well as Sobolev spaces. The following presentation is of reference character, focusing on the development of key concepts and results, omitting proofs where possible. We also give references for further studies of the respective issues.


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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Kristian Bredies
    • 1
  • Dirk Lorenz
    • 2
  1. 1.Institute for Mathematics and ScientificUniversity of GrazGrazAustria
  2. 2.BraunschweigGermany

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