G-Frame and Riesz Sequences in Hilbert Spaces |
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Authors: | Yan Zhang |
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Affiliation: | School of Mathematics and Information Science, North Minzu University, Yinchuan, P. R. China |
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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. |
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Keywords: | G-frame g-frame sequence g-Riesz sequence Gram matrix synthesis operator |
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