G-Frame and Riesz Sequences in Hilbert Spaces. (18th August 2019)

• Record Type: Journal Article
• Authors: Zhang, Yan; Li, Yun-Zhang
• Is Part Of: Numerical functional analysis and optimization. Volume 40:Number 11(2019)
• Page Start: 1268
• Page End: 1290
• Publication Date: 2019-08-18
• Subjects: G-frame -- g-frame sequence -- g-Riesz sequence -- Gram matrix -- synthesis operator; 42C15 -- 41A58 -- 47B38; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2019.1599912
• Languages: English
• ISSNs: 0163-0563

Abstract

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.