Fundamental functions vanishing on a given set and division by functions

The space $\Psi_V$ of fundamental functions (a subspace of S) consisting of functions vanishing together with all their derivatives on a given closed set $V\subset R^n$ is constructed. Multipliers in $\Psi_V$ are described. In the space $\Psi_V$ is easily realized the division of unity by an infinitely differentiable function, “vanishing slowly” for approximation to its zero set, (in particular, by a polynomial). In the case of a cone $V$ in $R^n$, a description of the dual space $\Phi_V$ consisting of the Fourier preimages of functions of $\Psi_V$ is given. The problem of multipliers in $\Phi_V$ is discussed.