Vanishing Moment Dilations and Applications to Balanced and Graph Frames
Abstract
Given a matrix A∈ Mk× N(BbbF), we examine finite frames {fj}j=1N with vanishing A-moments and investigate their connections with the frame dilation property. We establish several equivalent characterizations for the existence of vanishing A−moment dilations, including conditions involving dual frames, operator ranges, and linear independence properties associated with the matrix A. In particular, we show that every frame with redundancy k admits a dual frame with vanishing A-moments for some totally nonsingular matrix A. A key takeaway of this is that by invoking such a dual frame as the encoding frame, any frame with redundancy k (erasure robustness is not needed) allows perfect signal reconstructions for any number of ℓ-erasures (ℓ ≤ k) which only involves inverting a matrix of small size (ℓ) instead of inverting the subframe operator which is a matrix of large size (N−ℓ). A special case of this type of frames are k-th vanishing moment frames which have natural connections with graph frames. We characterize the existence of dual frames with k−th vanishing moments through polynomial cancellation identities and operator theoretical conditions. We further discuss some applications to problems in erasure recovery and graph frames. These results provide a unified framework linking vanishing moment conditions, dual frame theory, dilation theory, and graph theoretic constructions of frames.
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BibTeX
@misc{han2026vanishing,
title = {{Vanishing Moment Dilations and Applications to Balanced and Graph Frames}},
author = {Deguang Han and Shankhadeep Mondal and Kai Liu and Omar Nour},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11043},
eprint = {2610.11043},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11043},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Vanishing Moment Dilations and Applications to Balanced and Graph Frames
AU - Deguang Han
AU - Shankhadeep Mondal
AU - Kai Liu
AU - Omar Nour
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11043
UR - https://arxiv.org/abs/2610.11043
AB - Given a matrix A∈ M_k× N(BbbF), we examine finite frames {f_j}_j=1^N with vanishing A-moments and investigate their connections with the frame dilation property. We establish several equivalent characterizations for the existence of vanishing A−moment dilations, including conditions involving dual frames, operator ranges, and linear independence properties associated with the matrix A. In particular, we show that every frame with redundancy k admits a dual frame with vanishing A-moments for some totally nonsingular matrix A. A key takeaway of this is that by invoking such a dual frame as the encoding frame, any frame with redundancy k (erasure robustness is not needed) allows perfect signal reconstructions for any number of ℓ-erasures (ℓ ≤ k) which only involves inverting a matrix of small size (ℓ) instead of inverting the subframe operator which is a matrix of large size (N−ℓ). A special case of this type of frames are k-th vanishing moment frames which have natural connections with graph frames. We characterize the existence of dual frames with k−th vanishing moments through polynomial cancellation identities and operator theoretical conditions. We further discuss some applications to problems in erasure recovery and graph frames. These results provide a unified framework linking vanishing moment conditions, dual frame theory, dilation theory, and graph theoretic constructions of frames.
ER -