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Keywords:
Fourier transform; Henstock-Kurzweil integral; bounded variation function
Fourier transform; Henstock-Kurzweil integral; bounded variation function
Summary:
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.
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