G-Frame and Riesz Sequences in Hilbert Spaces. (18th August 2019) - Zhang, Yan; Li, Yun-Zhang
Abstract: G -frames generalize frames in Hilbert spaces. The literatures show that g -frames and frames share many similar properties, while they behave differently in redundancy and perturbation properties. Interestingly, g -frames have been extensively studied, but g -frame sequences have not. This problem is nontrivial since a g -frame and a frame both involve all vectors in the same Hilbert space, while a g -frame sequence and a frame sequence do not. They involve different linear spans. Using the synthesis and Gram matrix methods, we in this paper characterize g -frame sequences and g -Riesz sequences; obtain the Pythagorean theorem for g -orthonormal systems. These results recover several known results and lead to some new results on g -frames.