5 - Buschman–Erdélyi integral and transmutation operators

Abstract

This chapter introduces an important class of transmutations, namely, Buschman–Erdélyi integral and transmutation operators. For example Riemann–Liouville fractional integrals, classical Sonine and Poisson operators, and Mehler–Fock transforms are special cases of this class. The name “Buschman–Erdélyi integral and transmutation operators” was introduced by S. M. Sitnik. We consider for Buschman–Erdélyi operators definitions and main properties, an exact and convenient classification, different factorizations and connections with the Mellin transform, norm estimates, and a criterion for unitarity for half-integer parameters. We also introduce new classes of Sonine–Katrakhov and Poisson–Katrakhov transmutations which are unitary for all real parameters and multi-dimensional versions. Different applications of Buschman–Erdélyi operators are considered in this chapter.