On the theorem of Frullani
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- by Juan Arias-de-Reyna PDF
- Proc. Amer. Math. Soc. 109 (1990), 165-175 Request permission
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 \[ \int _0^\infty {\frac {{f(ax) - f(bx)}}{x}} dx = Alog \frac {a}{b}.\] 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.References
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Additional Information
- © Copyright 1990 American Mathematical Society
- 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