Division by a Polynomial in Spaces of Analytic Functionals

We prove the unique solvability of the problem about division by a polynomial in separated variables under additional conditions on the quotient in the space of analytic functionals and obtain a representation of the solution satisfying zero conditions in the form of a generalized Duhamel product. The results obtained for the space of analytic functionals are applied to the space of functions that are holomorphic in convex polystarshaped domains with respect to the points 0 in ℂ^N and a partial differential equation, the space of all entire functions in ℂ^N and an equation with the Gelfond–Leont’ev derivatives, the spaces of infinitely differentiable and ultradifferentiable functions in a convex polycylindrical domain in ℝ^N and a partial differential equation in these spaces.