Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Dual Frame Completion Problem

Roza Aceska, Yeon Hyang Kim, Sivaram K. Narayan

TL;DR

The paper investigates exact dual-frame completion for a finite frame $F$ in $\mathbb{F}^n$ by asking when a dual frame $G$ can be chosen to include a given set of vectors $H$ as its first $s$ elements. It develops three constructive pathways—direct, indirect via a product-matrix representation, and via singular value decomposition (SVD)—to characterize existence and to produce explicit duals, with clear criteria for uniqueness and multiplicity based on $s$ and the span properties of subframes. The results provide both practical algorithms (pseudoinverse-based and closed-form formulas) and theoretical insight into how prescribed dual components constrain the remaining dual vectors, enabling erasure-resilient reconstruction and flexible dual-frame design. The work thus broadens the toolbox for frame design under fixed-dual constraints and connects classical dual-frame theory to modern reconstruction scenarios.

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 $F$.

Dual Frame Completion Problem

TL;DR

The paper investigates exact dual-frame completion for a finite frame in by asking when a dual frame can be chosen to include a given set of vectors as its first elements. It develops three constructive pathways—direct, indirect via a product-matrix representation, and via singular value decomposition (SVD)—to characterize existence and to produce explicit duals, with clear criteria for uniqueness and multiplicity based on and the span properties of subframes. The results provide both practical algorithms (pseudoinverse-based and closed-form formulas) and theoretical insight into how prescribed dual components constrain the remaining dual vectors, enabling erasure-resilient reconstruction and flexible dual-frame design. The work thus broadens the toolbox for frame design under fixed-dual constraints and connects classical dual-frame theory to modern reconstruction scenarios.

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 for a finite dimensional Hilbert space, and a set of vectors that is assumed to be a subset of a dual frame of , we answer the following question: Which dual frame for - if it exists - completes the given set ? Solutions are explored through a direct and an indirect approach, as well as via the singular value decomposition of the synthesis operator of .
Paper Structure (6 sections, 9 theorems, 43 equations)

This paper contains 6 sections, 9 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A surgery problem in finding the dual frame
  4. On the direct approach
  5. On the indirect approach - via product matrices
  6. The Singular Value Decomposition Approach

Key Result

Theorem 4

Let $A$, $B$ and $C$ be some matrices such that $AB=C$. The following statements are equivalent: If $A$ has full row rank, then the pseudoinverse $A^+$ is given by and a particular solution $B$ can be found using $A^+$ as $B = A^+ C.$

Theorems & Definitions (31)

  • Definition 1
  • Definition 3
  • Theorem 4
  • Example 5
  • Theorem 6
  • proof
  • Corollary 7
  • proof
  • Theorem 8
  • proof
  • ...and 21 more