Two-component window system based on coherent states and theta functions

In this paper, we construct a two-component window system of functions with good time-frequency localization. The system consists of two window subfamilies orthogonal to each other. The procedure for orthogonalizing the resulting subfamilies is discussed, explicit formulas for calculating the uncertainty constants are given, and the problem of completeness of the whole two-component system is considered. Questions about orthogonalization and completeness are reduced to testing a certain hypothesis about the zeros of the Zak transform.