A necessary condition for HK-integrability of the Fourier sine transform function

Juan H. Arredondo; Manuel Bernal; Maria G. Morales

Czechoslovak Mathematical Journal (2025)

  • Issue: 1, page 69-84
  • ISSN: 0011-4642

Abstract

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The paper is concerned with integrability of the Fourier sine transform function when f ∈ BV 0 ( ℝ ) , where BV 0 ( ℝ ) is the space of bounded variation functions vanishing at infinity. It is shown that for the Fourier sine transform function of f to be integrable in the Henstock-Kurzweil sense, it is necessary that f / x ∈ L 1 ( ℝ ) . We prove that this condition is optimal through the theoretical scope of the Henstock-Kurzweil integration theory.

How to cite

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Arredondo, Juan H., Bernal, Manuel, and Morales, Maria G.. "A necessary condition for HK-integrability of the Fourier sine transform function." Czechoslovak Mathematical Journal (2025): 69-84. <http://eudml.org/doc/299918>.

@article{Arredondo2025,
abstract = {The paper is concerned with integrability of the Fourier sine transform function when $f\in \{\rm BV\}_0(\mathbb \{R\} )$, where $\{\rm BV\}_0(\mathbb \{R\} )$ is the space of bounded variation functions vanishing at infinity. It is shown that for the Fourier sine transform function of $f$ to be integrable in the Henstock-Kurzweil sense, it is necessary that $f /x \in L^1(\mathbb \{R\})$. We prove that this condition is optimal through the theoretical scope of the Henstock-Kurzweil integration theory.},
author = {Arredondo, Juan H., Bernal, Manuel, Morales, Maria G.},
journal = {Czechoslovak Mathematical Journal},
keywords = {Fourier transform; Henstock-Kurzweil integral; bounded variation function},
language = {eng},
number = {1},
pages = {69-84},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A necessary condition for HK-integrability of the Fourier sine transform function},
url = {http://eudml.org/doc/299918},
year = {2025},
}

TY - JOUR
AU - Arredondo, Juan H.
AU - Bernal, Manuel
AU - Morales, Maria G.
TI - A necessary condition for HK-integrability of the Fourier sine transform function
JO - Czechoslovak Mathematical Journal
PY - 2025
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
IS - 1
SP - 69
EP - 84
AB - The paper is concerned with integrability of the Fourier sine transform function when $f\in {\rm BV}_0(\mathbb {R} )$, where ${\rm BV}_0(\mathbb {R} )$ is the space of bounded variation functions vanishing at infinity. It is shown that for the Fourier sine transform function of $f$ to be integrable in the Henstock-Kurzweil sense, it is necessary that $f /x \in L^1(\mathbb {R})$. We prove that this condition is optimal through the theoretical scope of the Henstock-Kurzweil integration theory.
LA - eng
KW - Fourier transform; Henstock-Kurzweil integral; bounded variation function
UR - http://eudml.org/doc/299918
ER -

References

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