Different analytic methods (perturbation theory in the Born approximation and under Bragg reflection, as well as coupled-wave theory and its modifications) are used to derive and discuss approximate analytic expressions for electromagnetic wave fields in bounded one-dimensional periodic dielectric structures and the corresponding reflection coefficients. The range of validity of each of the analytic solutions is established and it is shown that the modified coupled-wave method, which is valid simultaneously for large and small modulation periods and appreciable modulation depths, has the widest range of validity. The method is used to calculate the reflection coefficients of such structures as functions of the incident-wave frequency, taking into account the finite size of the structures, the properties of the ambient media, absorption, and small nonlinearity and aperiodicity.