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Fourier Analysis with Generalized Integration

Author

Listed:
  • Juan H. Arredondo

    (Departamento de Matemáticas, Universidad Autónoma Metropolitana—Iztapalapa, Av. San Rafael Atlixco 186, México City 09340, Mexico
    All authors contributed equally to this work.)

  • Manuel Bernal

    (Departamento de Matemáticas, Universidad Autónoma Metropolitana—Iztapalapa, Av. San Rafael Atlixco 186, México City 09340, Mexico
    All authors contributed equally to this work.)

  • María Guadalupe Morales

    (Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Kotlářská 2, 611 37 Brno, Czech Republic
    All authors contributed equally to this work.)

Abstract

We generalize the classic Fourier transform operator F p by using the Henstock–Kurzweil integral theory. It is shown that the operator equals the H K -Fourier transform on a dense subspace of L p , 1 < p ≤ 2 . In particular, a theoretical scope of this representation is raised to approximate the Fourier transform of functions on the mentioned subspace numerically. Besides, we show the differentiability of the Fourier transform function F p ( f ) under more general conditions than in Lebesgue’s theory. Additionally, continuity of the Fourier Sine transform operator into the space of Henstock-Kurzweil integrable functions is proved, which is similar in spirit to the already known result for the Fourier Cosine transform operator. Because our results establish a representation of the Fourier transform with more properties than in Lebesgue’s theory, these results might contribute to development of better algorithms of numerical integration, which are very important in applications.

Suggested Citation

  • Juan H. Arredondo & Manuel Bernal & María Guadalupe Morales, 2020. "Fourier Analysis with Generalized Integration," Mathematics, MDPI, vol. 8(7), pages 1-16, July.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:7:p:1199-:d:387741
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    References listed on IDEAS

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    1. Salvador Sánchez-Perales & Francisco J. Mendoza Torres & Juan A. Escamilla Reyna, 2012. "Henstock-Kurzweil Integral Transforms," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2012, pages 1-11, October.
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