arXiv subjects
Publications and source records attributed to Lalit K. Vashisht.
In this paper, we present necessary and sufficient conditions for some types of linear combination of frame elements (wave packet) to be a frame for $L^2(\mathbb{K})$, where $\mathbb{K}$ is a local field of positive characteristic.
In this paper we study frame-like properties of a wave packet system by using hyponormal operators on $L^2(\mathbb{R})$. We present necessary and sufficient conditions in terms of relative hyponormality of operators for a system to be a wave packet frame in $L^2(\mathbb{R})$. A characterization of hyponormal operators by using tight wave packet frames is proved. This is different from a method proved by Djordjevi$\acute{c}$ by using the Moore-Penrose inverse of a bounded linear operator with a closed range. The linear combinations of wave packet frames generated by hyponormal operators are discussed.
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 $\{ϕ_{i}\}_{i \in I}$ and $\{ψ_{i}\}_{i \in I}$ for a separable Hilbert space $\mathcal{H}$ are woven if there are positive constants $A \leq B$ such that for every subset $σ\subset I$, the family $\{ϕ_{i}\}_{i \in σ} \cup \{ψ_{i}\}_{i \in σ^{c}}$ 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.