On applications of the dihedral group to interpolation problems for entire functions

We consider a particular case of the dihedral group of rotations and study linear poly-element functional equations associated with that group. We search for a solution in the class of functions that are holomorphic in the plane with a cut along “half” of the boundary of its fundamental region and vanish at infinity. We suggest a method for the regularization of such equations based on the theory of the Carleman boundary-value problem. The inverse involutive shift is induced by the generating transformations of the group. The solution is searched in the form of a Cauchy-type integral with an unknown density. The solution is a lower function that is Borel-associated with a certain entire function of exponential type (upper function).