Phase and Norm Retrieval via Projections in 2-Inner Product Spaces

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DOI: https://doi.org/10.28924/2291-8639-24-2026-44
Published: Feb 19, 2026
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Salah H. Alshabhi, Phase and Norm Retrieval via Projections in 2-Inner Product Spaces, Int. J. Anal. Appl., 24 (2026), 44.

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Salah H. Alshabhi

Abstract

This paper introduces and studies 2-phase retrieval and 2-norm retrieval in the context of 2-inner product spaces, generalizing classical phase and norm retrieval problems to a nonlinear geometric setting. A collection of subspaces \(\{W_i\}_{i=1}^M\) in a 2-inner product space \(V\) is said to yield {2-phase retrieval} if the 2-norms of projections \(\|P_i s\|_z = \|P_i b\|_z\) for all \(i\) and all reference vectors \(z \in V \setminus \{0\}\) imply that \(s\) and \(b\) are phase-equivalent (i.e., \(s = c b\) for some scalar \(c\) with \(|c| = 1\)). Similarly, \(\{W_i\}_{i=1}^M\) achieves 2-norm retrieval if the 2-norms \(\|P_i s\|_z\) uniquely determine the 2-norm \(\|s\|_z\) for all \(s \in V\) and all \(z\).

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