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Numerical Applications of DFT

  • Gerlind Plonka
  • Daniel Potts
  • Gabriele Steidl
  • Manfred Tasche
Chapter
First Online:
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

This chapter addresses numerical applications of DFTs. In Sect. 9.1, we describe a powerful multidimensional approximation method, the so-called cardinal interpolation by translates φ(⋅ −k) with \(\mathbf k \in {\mathbb Z}^d\), where \(\varphi \in C_c(\mathbb R^d)\) is a compactly supported, continuous function. In this approximation method, the cardinal Lagrange function is of main interest. Applying this technique, we compute the multidimensional Fourier transform by the method of attenuation factors. Then, in Sect. 9.2, we investigate the periodic interpolation by translates on a uniform mesh, where we use the close connection between periodic and cardinal interpolation by translates. The central notion is the periodic Lagrange function. Using the periodic Lagrange function, we calculate the Fourier coefficients of a multivariate periodic function by the method of attenuation factors.

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Gerlind Plonka
    • 1
  • Daniel Potts
    • 2
  • Gabriele Steidl
    • 3
  • Manfred Tasche
    • 4
  1. 1.University of GöttingenGöttingenGermany
  2. 2.Chemnitz University of TechnologyChemnitzGermany
  3. 3.TU KaiserslauternKaiserslauternGermany
  4. 4.University of RostockRostockGermany

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