The frame for algorithm signal recovery

A frame of a finite-dimensional Euclidean space composed of vectors and their sums is considered. The operator proof of frame properties of the constructed system is presented, the eigenvalues for the matrix of the frame operator are found, which are also frame boundaries. The property of alternative completeness of the constructed system is proved. This property is the cause of interest in the constructed frame, since in real Euclidean space it is equivalent to injectivity of the measurement operator, which maps the signal vector into a sequence of measurement modules. The investigated frame underlies the fast algorithm of signal reconstruction, proposed by M. Shapiro and stated in [1]. An operator that translates the constructed frame into the Parseval–Steklov frame closest to it, is found.