Discretization of continuous frame

TL;DR

This paper introduces a new concept called continuous atomic resolution of identity for Hilbert spaces, providing a framework for continuous frames with countable reconstruction and analyzing their stability under perturbations.

Contribution

It defines the novel concept of continuous atomic resolution of identity and explores its relationship with existing continuous frames, including stability properties.

Findings

Characterization of the relationship between the new concept and known continuous frames

Establishment of stability under perturbations

Provision of a countable reconstruction formula for the new framework

Abstract

In this paper we consider on the notion of continuous frame of subspace and define a new concept of continuous frame, entitled {\it continuous atomic resolution of identity}, for arbitrary Hilbert space $\h$ which has a countable reconstruction formula. Among the other result, we characterize the relationship between this new concept and other known continuous frame. Finally, we state and prove the assertions of the stability of perturbation in this concept.