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Mellin transform analysis and integration by parts for Hadamard-type fractional integrals. (English) Zbl 1022.26011
The authors consider the known construction of Hadamard fractional integration \[ \mathcal{I}^\alpha_{0+,\mu} f(x)= \frac{1}{\Gamma(\alpha)}\int_0^x\left(\frac{u}{x}\right) ^\mu \left(ln \frac{x}{u}\right)^{ \alpha -1}\frac{f(u) du}{u} \] and some of their modifications. These constructions are invariant with respect to dilations and are related to the Liouville form of fractional integration via the corresponding change of variables. They study the Mellin transforms of \(\mathcal{I}^\alpha_{0+,\mu} f(x)\) when \(f\) is in the Lebesgue space with a power weight and obtain relations of the type of fractional integration by parts.

26A33 Fractional derivatives and integrals
44A15 Special integral transforms (Legendre, Hilbert, etc.)
Full Text: DOI
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