Characterizations and redundancies of g-frames in Hilbert spaces

Abstract

Let $\{{\mathrm{\Lambda }}_j:j\in J\}$ be a g-Bessel sequence of a Hilbert space. This study uses the type-II induced sequences $\{{\mathrm{\Gamma }}_{jk}{\mathrm{\Lambda }}_j:j\in J,k\in K_j\}$ of $\{{\mathrm{\Lambda }}_j:j\in J\}$ to characterize $\{{\mathrm{\Lambda }}_j:j\in J\}$ to be a tight g-frame and a g-orthonormal basis. We also obtain the exact relationship between the synthesis operators of $\{{\mathrm{\Lambda }}_j:j\in J\}$ and its type-II induced sequences. We then use the type-I induced sequences of $\{{\mathrm{\Lambda }}_j:j\in J\}$ to characterize the (maximum) robustness to erasures and (maximum) uniform excess of $\{{\mathrm{\Lambda }}_j:j\in J\}$, and vice versa. Finally, we estimate the upper error bound of type-I induced sequences under some perturbations of $\{{\mathrm{\Lambda }}_j:j\in J\}$ and $\{{\mathrm{\Gamma }}_j:j\in J\}$, and vice versa.

Keywords:

• G-frame
• g-orthonormal basis
• induced sequence
• erasure
• uniform excess

2010 Mathematics Subject Classifications:

• 42C15
• 42C40

Disclosure statement

The authors declare that they have no conflict of interest.

Additional information

Funding

This work is partly supported by the Natural Science Foundation of Fujian Province, China (Grant Nos. 2020J01267 and 2021J011192), and the Projects of Xiamen University of Technology (Grant Nos. 40199071 and 50419004).