Fractional Fourier transform in the framework of fractional calculus operators

Abstract The paper is devoted to the study of a fractional Fourier transform in a special space of functions by Lizorkin. Connections of such a transform with differentiation operators are established and an inverse operator for such a transform if constructed. Compositions of the fractional Fourier transform with modified fractional integrals and derivatives are proved. Application to solution of a partial diffusion-type differential equation of fractional order is given. Keywords: Fourier and fractional Fourier transformsmodified Liouville fractional derivatives and integralsoperational relationsdiffusion-type fractional differential equationLaplace transformMittag–Leffler function 2000 MSC Subject Classification Codes : 42A3826A3335A2235L1544A1047G1047B3833E12 Acknowledgements The project work was started during the visit of the first author to the University of La Laguna (ULL) in November of 2009. This work was supported, in part, by MEC (grant MTM2007-60246), by ULL and by the Belarusian Fundamental Research Fund (Project F08MC-028), Additional informationNotes on contributorsA. A. Kilbas A.A. Kilbas passed away in June, 2010

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