Stationarity of the Gabor basis and derivation of Janssen's formula

The theory of stationary processes is applied to homogeneous functional sequences. Examples of time and frequency shifts and dilations are considered. It is proved that the functional sequences are homogeneous (stationary) and therefore correspond to a one-dimensional stationary vector field. The spectral properties of these sequences are explored. It is also shown that the Gabor expansion corresponds to a two-dimensional homogeneous vector field. The spectral properties of the Gabor (1946) expansion are considered, and the formula for the coefficients of the Gabor decomposition is derived.<>