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2 results

  • 13 - Trigonometric Interpolation and the Fast Fourier Transform

  • from Part II - Constructive Approximation Theory
  • Abner J. Salgado, University of Tennessee, Knoxville, Steven M. Wise, University of Tennessee, Knoxville
  • Book:
    Classical Numerical Analysis
    Published online:
    29 September 2022
    Print publication:
    20 October 2022, pp 345-371

  • Summary

    This chapter begins with the study of trigonometric interpolation and the discrete Fourier transform. As a first application, the numerical integration of periodic functions is discussed. More detailed topics, like existence and uniqueness of trigonometric interpolants, as well as alias as convergence of trigonometric interpolation are discussed. An important practical tool, the fast Fourier transform (FFT) is then introduced. With this at hand the most rudimentary ideas of signal processing are presented.

  • 4 - Unipotent Characters

  • Meinolf Geck, Universität Stuttgart, Gunter Malle, Technische Universität Kaiserslautern, Germany
  • Book:
    The Character Theory of Finite Groups of Lie Type
    Published online:
    20 February 2020
    Print publication:
    27 February 2020, pp 271-362

  • Summary

    After collectiong some properties of irreducible representations of finite Coxeter groups we state and explain Lusztig‘s result on the decomposition of Deligne-Lusztig characters and then givea detailed exposition of the parametrisation and the properties of unipotent characters of finite reductive groups and related data like Fourier matrices and eigenvalues of Frobenius. We then describe the decomposition of Lusztig induction and collect the most recent results on its commutation with Jordan decomposition. We end the chapter with a survey of the character theory of finite disconnected reductive groups.