Continuous k-frames and their duals

TL;DR

This paper introduces continuous k-frames (ck-frames) in Hilbert spaces, constructs duals for them, and explores their properties, extending the theory of frames with a focus on atomic systems and continuous representations.

Contribution

It presents the first construction of ck-frames, introduces a novel method for duals, and advances the understanding of continuous frames related to bounded linear operators.

Findings

Construction of continuous k-frames (ck-frames) in Hilbert spaces.

A new method for obtaining duals of ck-frames.

Proved new theoretical results about ck-frames and their duals.

Abstract

k-frames were recently introduced by Gavruta in Hilbert spaces to study atomic systems with respect to a bounded linear operator. A continuous frame is a family of vectors in Hilbert space which allows reproductions of arbitrary elements by continuous super positions. We construct a continuous k-frame, so called ck-frame along with an atomic system for this version of frames. Also we introduce a new method for obtaining the dual of a ck-frame and we prove some new results about it.