We have numerically studied the band structure and the spin polarization effect in a periodic one-dimensional array of quantum point contacts (QPCs) formed in a two-dimensional electron gas in a plane-layered semiconductor system. In this study we used a self-consistent model developed within the framework of the Kohn-Sham local spin-density formalism. We have found that the band structure contains a mixture of flat and dispersed bands, and the role of transverse modes in the formation of such a band structure has been clearly demonstrated. We have also shown that spin polarization occurs mainly in the regions occupied by the QPCs and that it is qualitatively similar to the spin polarization in a single QPC.