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On the theorem of Frullani


Author: Juan Arias-de-Reyna
Journal: Proc. Amer. Math. Soc. 109 (1990), 165-175
MSC: Primary 26A39; Secondary 42A38
DOI: https://doi.org/10.1090/S0002-9939-1990-1007485-4
MathSciNet review: 1007485
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Abstract: We prove that, for every function $ f:{{\mathbf{R}}^ + } \to {\text{C}}$ such that $ (f(ax) - f(bx))/x$ is Denjoy-Perron integrable on $ [0, + \infty )$ for every pair of positive real numbers $ a,b$, there exists a constant $ A$ (depending only on the values of $ f(t)$ in the neighborhood of 0 and $ + \infty $) such that

$\displaystyle \int_0^\infty {\frac{{f(ax) - f(bx)}}{x}} dx = Alog \frac{a}{b}.$

To prove this assertion, we identify a Denjoy-Perron integrable function $ f:{\mathbf{R}} \to {\text{C}}$ with a distribution. In this way, we obtain the main result of this paper: The value at 0 (in Lojasiewicz sense) of the Fourier transform of the distribution $ f$ is the Denjoy-Perron integral of $ f$. Assuming the Continuum Hypothesis, we construct an example of a non-Lebesgue measurable function that satisfies the hypotheses of the first theorem.

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DOI: https://doi.org/10.1090/S0002-9939-1990-1007485-4
Article copyright: © Copyright 1990 American Mathematical Society

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