On Description of Dual Frames

TL;DR

This paper presents a comprehensive method for characterizing all dual frames in signal reconstruction, utilizing Naimark's dilation theorem, with simplified formulas for finite-dimensional spaces and near-Riesz bases.

Contribution

It introduces a novel approach to find every dual frame using Naimark's dilation theorem, extending the understanding of frame duality in various settings.

Findings

Provides a general formula for dual frames

Simplifies formulas for finite-dimensional spaces

Extends results to near-Riesz bases

Abstract

One of a key problems in signal reconstruction process with the use of frames is to find a dual frame. Typically, a canonical dual frame is used. However, there are many applications where this choice appears to be unfortunate. Due to that fact, it is necessary to develop a tool, which helps to find a suitable dual frame. In this paper we give a method to find every dual frames. The proposed method is based on Naimark's dilation theorem and the obtained description of dual frames involves parameters that characterize extension of a Parseval frame to an orthonormal basis. These formulas are simplified for frames in finite-dimensional spaces and for near-Riesz bases. In the latter case, the simplification is based on the extended and supplemented version of the Naimark theorem, which is proved in the last part of the paper.