This work is concerned with density of some rather common classes of entire functions (for example, with density of the class of all entire functions of order less than $\sigma$) in $\Phi$-spaces. One of the typical examples of а $\Phi$-space is $C(E)$, where $E$ is a closed subset of $\mathbb{R}$. The dual problems reduce to questions concerning the quasianalyticity of classes of functions with Fourier transforms supported on a thin set. The latter problems are treated as problems of potential theory. Conformal mappings of the upper-half plane to special "comb-like" domains are systematically used.