On certain one- and two-dimensional hypersingular integral equations

For two-dimensional singular and hypersingular integrals more general definitions than the traditional ones are introduced. For a hypersingular operator on a sphere a new spectral relation is obtained. Quadrature formulae of the kind of discrete vortex pairs for one-dimensional and of the kind of closed vortex frames for two-dimensional hypersingular integrals are considered; questions on their convergence are discussed, as well as the convergence of numerical solutions to the corresponding hypersingular integral equations on a finite line interval and a circle. An experiment on the numerical solution of a hypersingular integral equation on a sphere is carried out, which demonstrates analogies between numerical solutions of hypersingular integral equations on a finite interval and a sphere.