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The Ordinary Discrete Fourier Transform

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The XFT Quadrature in Discrete Fourier Analysis

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

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Abstract

This chapter is intended to complete the structure of this book. We present in this chapter an overview of the usual discretization of the Fourier transform as given by the discrete Fourier transform (DFT). Some little-known properties and drawbacks of this discretization are shown in this chapter. The cases of discrete rotations and translations are discussed and an unusual application of these operators is given. A fractional partial differential equation for theta functions and a numerical procedure for computing fractional partial derivatives of theta functions and elliptic integrals are presented as new results of this approach.

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Notes

  1. 1.

    For example, the function 1∕(1 + |t|) belongs to L2(−∞, ∞) but not to L1(−∞, ∞). On the other side, the function \(1/\sqrt {\vert t\vert }\) for |t| < 1, 0 otherwise, belongs to L1(−∞, ∞) but not to L2(−∞, ∞).

  2. 2.

    It will be shown in Sect. 2.3.1 that this formalism also holds for N even.

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Authors and Affiliations

  1. Science Department, University of Quintana Roo, Chetumal, Mexico

    Rafael G. Campos

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  1. Rafael G. Campos
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Campos, R.G. (2019). The Ordinary Discrete Fourier Transform. In: The XFT Quadrature in Discrete Fourier Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-13423-5_2

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