Weaving K-frames in Hilbert Spaces

Abstract

Gavruta introduced $K$-frames for Hilbert spaces to study atomic systems with respect to a bounded linear operator. There are many differences between K-frames and standard frames, so we study weaving properties of K-frames. Two frames $\{{\varphi }_i{\}}_{i\in I}$ and $\{{\psi }_i{\}}_{i\in I}$ for a separable Hilbert space $\mathcal{H}$ are woven if there are positive constants $A\le B$ such that for every subset $\sigma \subset I$, the family $\{{\varphi }_i{\}}_{i\in \sigma }\cup \{{\psi }_i{\}}_{i\in {\sigma }^c}$ is a frame for $\mathcal{H}$ with frame bounds $A,B$. In this paper, we present necessary and sufficient conditions for weaving $K$-frames in Hilbert spaces. It is shown that woven $K$-frames and weakly woven $K$-frames are equivalent. Finally, sufficient conditions for Paley-Wiener type perturbation of weaving $K$-frames are given.

References in corpus (1)

• Multi-Window Weaving Frames

Cited by in corpus (2)

• Vector-Valued (Super) Weaving Frames
• Operators for matrix-valued Riesz bases over LCA groups