Equivalence relations and distances between generalized frames

• Published in: Journal of pseudo-differential operators and applications Vol. 13; no. 4
• Main Author: Deepshikha
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.12.2022; Springer Nature B.V
• Subjects: Algebra; Analysis; Applications of Mathematics; Equivalence; Frames; Functional Analysis; Mathematics; Mathematics and Statistics; Operator Theory; Partial Differential Equations; Projectors; Subspaces; Trigonometric functions; Upper bounds; 46C05; 42C40; Closeness bound; 42C30; Distance between frames; 42C15; Generalized frames; Nearness
• ISSN: 1662-9981; 1662-999X
• DOI: 10.1007/s11868-022-00479-2

G -frames (or generalized frames) are natural generalizations of frames including ordinary frames, frames of subspaces, bounded quasi-projectors, pseudo-frames, etc. In this paper, we study some properties of equivalence relations between generalized frames. We characterize equivalent g -frames and near g -frames in terms of equivalent ordinary frames and near ordinary frames, respectively. Further, an upper bound for the closeness bound is presented. We prove that g -frames are near if and only if their respective canonical dual g -frames are near. It is also shown that any two generalized orthonormal bases (or generalized Riesz bases) are near. Finally, we define a metric on near g-frames using inverse trigonometric function and find the closest and nearest tight g -frame to a given g -frame.