Spectral synthesis for an operator generated by multiplication by a power of the independent variable

This paper is devoted to spectral synthesis of the operator adjoint to multiplication by a power of the independent variable in weighted spaces of entire functions of a single complex variable, and is closely connected with equations of convolution type, and with the general theory of subspaces invariant under a multiple differentiation operator. The problem of approximating solutions of a homogeneous equation of $q$-sided convolution type by elementary solutions is solved. Certain systems of such equations are investigated.