Symbolic calculation and invertibility of convolution operators on the infinite dihedral group

Nowadays, convolution operators on discrete noncommutative groups are under intensive research due to their applications, in particular, in the theory and practice of data networking, in image analysis, and in problems of diffraction by bodies with a noncommutative symmetry group. The symbolic calculation for algebra of convolution equations on the noncommutative infinite dihedral group $\mathbb{D}_\infty$ has been developed. Necessary and sufficient conditions of invertibility of convolution operators from this algebra in terms of symbolic calculation have been found in this paper. Besides, inclosure of algebra of convolution equations on $\mathbb{D}_\infty$ into matrix algebra of convolution operators on the group of whole numbers extended with involutive operator has been constructed.
In the theory of projection methods of the solution of operator equations the sequence of equations with more simple operators is constructed in order to approximate the solution of original equation with some accuracy, i.e. the reduction of original invertible operator to a more simple invertible operator. The connection between dual object of $\mathbb{D}_\infty$ and finite noncommutative dihedral group $\mathbb{D}_m$ is studied. On the basis of this the operator of reduction that maps invertible operator of convolution on $\mathbb{D}_\infty$ to invertible convolution operator on $\mathbb{D}_m$ is constructed in this paper.