Two Controlled Frames in Hilbert Spaces

One of the last extensions of frames is weighted and controlled frames that introduced by P.Balazs J-P. Antoine and A. Grybos to improve the numerical efficiency of iterative algorithms for inverting the frame operator. It is natural to ask: how we can apply two controlling operator on a family of vectors and build a frame with special controlling properties? In this manuscript, we define the concept of two controlling frame and we prove some of its properties. We will show these frames give a new approach for approximation of elements in H. Our result can be extended to finitely many controlling frames.