About an existence theorem of the Henstock-Fourier transform
DOI:
https://doi.org/10.4067/S0716-09172008000300006Keywords:
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.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).
[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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