Abstract
The so-called Chen modification of the Liouville fractional integrals (LFI) allows to study LFI of functions which may have arbitrary behaviour at both −∞ and +∞. We develop a similar approach for dilation invariant Hadamard fractional integro-differentiation on \( {\mathbb{R}_ +} \). We introduce several types of truncation of the corresponding Marchaud form of fractional Chen– Hadamard fractional derivatives and show that these truncations applied to Chen–Hadamard fractional integral of a function f in \( L_{loc}^p({\mathbb{R}_ + })\,or\,{L^p}({\mathbb{R}_ + }) \) converge to this function in Lp-norm, locally or globally, respectively. In the local case, we admit functions f with an arbitrary growth both at the origin and infinity.
Mathematics Subject Classification (2010). Primary 26A33.
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Dedicated to Professor António Ferreira dos Santos
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Samko, S., Yakhshiboev, M.U. (2014). A Chen-type Modification of Hadamard Fractional Integro-Differentiation. In: Bastos, M., Lebre, A., Samko, S., Spitkovsky, I. (eds) Operator Theory, Operator Algebras and Applications. Operator Theory: Advances and Applications, vol 242. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0816-3_20
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DOI: https://doi.org/10.1007/978-3-0348-0816-3_20
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