Dual Frame Completion Problem

TL;DR

This paper investigates the construction of an exact dual frame in finite-dimensional Hilbert spaces, given partial information, using direct, indirect, and SVD-based methods.

Contribution

It introduces methods to determine and construct dual frames that complete a given subset, under specific structural assumptions.

Findings

Provides conditions for the existence of a dual frame completing a given set.

Develops direct and indirect construction algorithms for dual frames.

Utilizes singular value decomposition to analyze and solve the completion problem.

Abstract

In this paper we present the construction of an exact dual frame under specific structural assumptions posed on the dual frame. When given a frame $F$ for a finite dimensional Hilbert space, and a set of vectors $H$ that is assumed to be a subset of a dual frame of $F$, we answer the following question: Which dual frame $G$ for $F$ - if it exists - completes the given set $H$? Solutions are explored through a direct and an indirect approach, as well as via the singular value decomposition of the synthesis operator of $F$.