# Characterization and Construction of K-Fusion Frames and Their Duals in Hilbert Spaces

## TL;DR

This paper introduces K-fusion frames in Hilbert spaces, providing methods for their identification, dual construction, and analyzing their robustness, with applications in sampling theory.

## Contribution

It defines K-fusion frames, offers new approaches for their characterization and dual construction, and studies their stability under perturbations.

## Key findings

- K-fusion frames can be characterized via multiple approaches.
- Explicit methods for constructing duals of K-fusion frames are provided.
- K-fusion frames exhibit robustness under certain perturbations.

## Abstract

K-frames, a new generalization of frames, were recently considered by L. Gavruta in connection with atomic systems and some problems arising in sampling theory. Also, fusion frames are an important generalization of frames, applied in a variety of applications. In the present paper, we introduce the notion of K-fusion frames in Hilbert spaces and obtain several approaches for identifying of $K$-fusion frames. The main purpose is to reconstruct the elements from the range of the bounded operator $K$ on a Hilbert space H by using a family of closed subspaces in H. This work will be useful in some problems in sampling theory which are processed by fusion frames. For this end, we present some descriptions for duality of K-fusion frames and also resolution of the operator K to provide simple and concrete constructions of duals of K-fusion frames. Finally, we survey the robustness of K-fusion frames under some perturbations.