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The Dirichlet-Jordan theorem for the Henstock-Fourier transform. (English) Zbl 1217.42016
The Dirichlet-Jordan theorem solves the pointwise inversion of the Fourier transform for Lebesgue integrable and bounded variation functions on $\Bbb R$. Using a Riemann-Lebesgue lemma for the Fourier transform over the class of bounded variation functions that vanish at infinity, the Dirichlet-Jordan theorem is proved for functions in this class. The proof given in this paper is in the Henstock-Kurzweil integral context and is different to that of Riesz-Livingston. As a consequence, the Dirichlet-Jordan theorem for functions in the intersection of the spaces of bounded variation functions and of Henstock-Kurzweil integrable functions is obtained. In this intersection there exist functions that are square integrable, but not Lebesgue integrable.
42A38Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
26A39Special integrals of functions of one real variable
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