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Zbl 1022.26011
Butzer, Paul L.; Kilbas, Anatoly A.; Trujillo, Juan J.
Mellin transform analysis and integration by parts for Hadamard-type fractional integrals.
(English)
[J] J. Math. Anal. Appl. 270, No.1, 1-15 (2002). ISSN 0022-247X

The authors consider the known construction of Hadamard fractional integration $$\cal{I}^\alpha_{0+,\mu} f(x)= \frac{1}{\Gamma(\alpha)}\int_0^x\left(\frac{u}{x}\right) ^\mu \left(ln \frac{x}{u}\right)^{ \alpha -1}\frac{f(u) du}{u}$$ 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 $\cal{I}^\alpha_{0+,\mu} f(x)$ when $f$ is in the Lebesgue space with a power weight and obtain relations of the type of fractional integration by parts.
[Stefan G.Samko (Faro)]
MSC 2000:
*26A33 Fractional derivatives and integrals (real functions)
44A15 Special transforms

Keywords: Hadamard-type fractional integration; Mellin transforms

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