Hypersingular Integrals of Marchaud Type and the Inversion Problem for Potentials

In 1927 A. MARCHAUD defined a fractional derivative of a function of one variable in the form of the integral containing the finite difference of this function. The purpose of the paper is to show that this idea can be generalized to become a foundation of the general method which enables to invert and to characterize a wide class of potential type operators with a semigroup property arising in analysis and in mathematical physics. This method leads to hypersingular integrals (HSI's), by means of which one can construct both explicit and stable approximate inverses to potentials. The paper contains the description and the justification of the method as well as its applications to various important one‐ and multidimensional potentials.