Czechoslovak Mathematical Journal, first online, pp. 1-16
A necessary condition for HK-integrability of the Fourier sine transform function
Juan H. Arredondo, Manuel Bernal, Maria G. Morales
Received June 22, 2022. Published online March 22, 2023.
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.
Keywords: Fourier transform; Henstock-Kurzweil integral; bounded variation function
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Affiliations: Juan H. Arredondo (corresponding author), Manuel Bernal, Department of Mathematics, Division of Basic Sciences and Engineering, Metropolitan Autonomous University, Av. San Rafael Atlixco 186, Del. Iztapalapa, C.P. 09310, Mexico City, Mexico, e-mail: iva@xanum.uam.mx, mbg@xanum.uam.mx; Maria G. Morales, Department of Mathematics and Statistics, Faculty of Science, Masaryk University. Kotlářská 2, 611 37 Brno, Czech Republic, e-mail: maciasm@math.muni.cz
