A Chen-type Modification of Hadamard Fractional Integro-Differentiation

Abstract

AbstractThe 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.KeywordsFractional integralfractional derivativeHadamard fractional integro-differentiationMarchaud fractional derivativestruncation of Marchaud derivatives