A method for the localization of singularities of a solution to a convolution-type equation of the first kind with a step kernel

We consider the problem on the localization of singularities (delta-functions) of a solution to a convolution-type equation of the first kind with a step kernel. We propose a new regularization method which allows to calculate the number of singularities and to approximate them. The accuracy of approximation is calculated. We obtain bounds for an important characteristic of the method, namely, the separability threshold. We prove the order-optimality of the proposed method on classes of functions with singularities both with respect to the accuracy and the separability.