About an existence theorem of the Henstock-Fourier transform

Authors

  • Francisco Javier Mendoza Torres Benemérita Universidad Autónoma de Puebla.
  • Juan Alberto Escamilla Reyna Benemérita Universidad Autónoma de Puebla.
  • María Guadalupe Raggi Cárdenas Benemérita Universidad Autónoma de Puebla.

DOI:

https://doi.org/10.4067/S0716-09172008000300006

Keywords:

Henstock-Kurzweil integral, bounded variation function, Lebesgue integral, integral de Henstock-Kurzweil, función de variación acotada, integral de Lebesgue.

Abstract

We show that if f is lying on the intersection of the space of Henstock-Kurzweil integrable functions and the space of the bounded variation functions in the neighborhood of ± ∞, then its Fourier Transform exists in all R. This result is more general than the classical result which enunciates that if f is Lebesgue integrable, then the Fourier Transform of f exists in all R, because we also have proved that there are functions which belong to the intersection of the space of the Henstock-Kurzweil integrable functions and the space of the bounded variation functions which are not Lebesgue integrable.

Author Biographies

Francisco Javier Mendoza Torres, Benemérita Universidad Autónoma de Puebla.

Facultad de Ciencias Físico Matemáticas.

Juan Alberto Escamilla Reyna, Benemérita Universidad Autónoma de Puebla.

Facultad de Ciencias Físico Matemáticas.

María Guadalupe Raggi Cárdenas, Benemérita Universidad Autónoma de Puebla.

Facultad de Ciencias Físico Matemáticas.

References

[1] Bartle, R.G., A Modern Theory of Integration, Graduate Studies in Mathematics, Vol 32, American Mathematical Society, Providence Rhode Island, (2001).

[2] Gordon, R. A., The Integral of Lebesgue, Denjoy, Perron, and Henstock, Graduate Studies in Mathematics, Vol 4, American Mathematical Society, Providence, (1994).

[3] Henstock, R., Lectures on The Theory of Integration, World Scientific Publications Co., Singapure, (1988).

[4] Lee Peng, Y., Lanzhou Lectures on Henstock Integration, Publications Co., Singapure, (1989).

[5] Y. E. Morales Rosado y F. J. Mendoza Torres, Algunos Aspectos de la Transformada de Fourier en el Espacio de las Funciones HK-Integrables, Aportaciones Matemáticas, Memorias SMM, 37, pp. 101-114, (2007).

[6] Talvila Erik, Henstock-Kurzweil Fourier Transforms, Illinois Journal of Mathematics, 46, pp. 1207-1226, (2002).

Published

2017-04-06

How to Cite

[1]
F. J. Mendoza Torres, J. A. Escamilla Reyna, and M. G. Raggi Cárdenas, “About an existence theorem of the Henstock-Fourier transform”, Proyecciones (Antofagasta, On line), vol. 27, no. 3, pp. 307-318, Apr. 2017.

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