Abstract
Using a Riemann-Lebesgue lemma for the Fourier transform over the class of bounded variation functions that vanish at infinity, we prove the Dirichlet--Jordan theorem for functions on this class. Our proof is in the Henstock--Kurzweil integral context and is different to that of Riesz-Livingston [Amer. Math. Monthly 62 (1955), 434--437]. As consequence, we obtain the Dirichlet--Jordan theorem for functions in the intersection of the spaces of bounded variation functions and of Henstock--Kurzweil integrable functions. In this intersection there exist functions in $L^{2}(\mathbb{R})\backslash L(\mathbb{% R}).$
Citation
Francisco J. Mendoza Torres. "On pointwise inversion of the Fourier transform of $BV_{0}$ functions." Ann. Funct. Anal. 1 (2) 112 - 120, 2010. https://doi.org/10.15352/afa/1399900593
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