Universal Symbols on Locally Compact Abelian Groups

From the viewpoint of elementary functional analysis, Bernstein inequalities are mainly sharp estimates for the norms of certain operators of convolution of entire functions bounded on the real line and having finite exponential type not exceeding a given one with (complex) Borel measures of finite total variation.
If we assume that the functions are defined on a locally compact Abelian group and use the uniform norms, then the generalized Bernstein spaces are parametrized by compact sets in the dual group $X$ and the symbols of the operators are the restrictions to compact sets in $X$ of functions locally coinciding with the Fourier transforms of measures. There exists symbols such that, in the case of uniform norms (and then, as it turns out, also in more general cases), the norm of the corresponding operator coincides with its spectral radius. The main result of the paper is a description of these (universal) symbols in terms of positive definite functions. Connected groups play a special role here.