Design and characterization of optimal dual frames under Hilbert-Schmidt norm
Optimization Eruditorum, Volume 2, Issue 3, December 2025, Pages 200–212
SHANKHADEEP MONDAL
Department of Mathematics, University of Central Florida, 4000 Central Florida Blvd, Orlando, FL 32816, USA
R. N. MOHAPATRA
Department of Mathematics, University of Central Florida, 4000 Central Florida Blvd, Orlando, FL 32816, USA
Abstract
This paper investigates the problem of identifying optimal dual frames in finite-dimensional Hilbert spaces when the reconstruction error, in the presence of erasures, is measured by the Hilbert–Schmidt norm. We present a unified framework for characterizing optimal duals for arbitrary $m$-erasure scenarios, with \(1 \leq m \leq N\), where \(N\) denotes the number of frame elements. Several results for canonical dual frame to be optimal are established, providing conditions under which the canonical dual is the unique optimal dual. We further construct explicit examples illustrating both the uniqueness and non-uniqueness phenomena, as well as situations where the canonical dual fails to be optimal. These results shed light on the structural properties of frames that govern erasure-robust reconstructions.
Cite this Article as
Shankhadeep Mondal and R. N. Mohapatra, Design and characterization of optimal dual frames under Hilbert-Schmidt norm, Optimization Eruditorum, 2(3), 200–212, 2025