[en] The equivalence of two types of triplets on the space of entire analytic functions is proved. The analytic expression for the Bargmann transformation of free field propagators in one-dimensional case with any value of the mass parameter is presented. In four-dimensional Minkowski space Bargmann transformations for massless particles is derived. In the case of massive particles it has been shown that the Bargmann transformations can be obtained in the form of power series representing entire functions. An analytic representation of the unitarity and causality conditions is given and the invariance of the growth and type with respect to Poincare transformations is shown. (author)