Minimal (non-polynomial) splines are constructed on a non-homogeneous meshes, the continuity condition for these splines and their derivatives in mesh points are formulated. The uniqueness conditions for linear spaces of smooth splines are given and their embedment is established for any sequence of refining meshes (at arbitrary refinement). Simple realizations of the system of functionals, biorthogonal to those corresponding to basis system, are constructed. As result, new wavelet decompositions are obtained, as well as direct solutions to interpolation problems in the spaces of polynomial and non-polynomial minimal splines. Theory is illustrated by examples of result application for m= 2.