Frames from Cosines with the Degenerate Coefficients

Abstract

The system of cosines with a degenerate coefficient in exponential form is considered. A necessary and sufficient condition on the degree of degeneration is found that makes the considered system a frame in Lebesgue spaces. It is proved that if the degenerate coefficient satisfies the Muckenhoupt condition, then the basicity holds. If the Muckenhoupt condition does not hold, then the system has a finite defect, and does not form a frame.