Generalized Frames and Riesz Systems

In this paper a generalization of frame systems is made. First description of systems of this type was made by T. P. Lukashenko. In 1997 he introduced a class of generalized similar to orthogonal systems, and in 2006 proposed an idea to expand of frame-based systems on the generalized space. This question is considered in this paper. Firstly, the paper gives the description of well-studied, as for now, discrete and integral frames, as well as describes the main practical applications of such frame systems. The paper considers generalized systems, similar to orthogonal, introduced by T. P. Lukashenko, and these systems are extended to generalized frames. Given examples indicate that an input class is more inclusive than previously considered discrete and integral frames, and more general than the generalized orthogonal system (examples are Fourier transformation and the Hilbert transformation). The concept of generalized Riesz system is introduced and the relationship between frames and Riesz systems in a generalized way is studied. Two theorems are proved in the work to establish close links between the introduced generalized frames and generalized Riesz systems. The theorem give the necessary and sufficient criteria for the system to be a generalized frame. Parseval’s identity analog is deduced for generalized frame systems.

exponential system, frames, Riesz system, generalized systems similar to orthogonal, orthogonal projections, Parseval’s identity