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Grassmannian frames offer redundant vector representations characterised by low mutual coherence, which makes them useful in communication, signal recovery, and erasure-resilient data reconstruction. A new geometric and operator-theoretic framework uses numerical range theory alongside complex projective geometry to evaluate finite-dimensional Grassmannian frames. In this setting, the numerical ranges of the frame operator and the Gramian operator are determined by their spectral extrema. Frame tightness corresponds directly to the numerical range of the frame operator collapsing to a single point. Connecting the numerical range to the Rayleigh map on complex projective space reveals that critical points represent eigenvectors of the frame operator. Additionally, the framework assesses erasure robustness using masking matrices and error operators, establishing lower bounds for remaining operators following multiple erasures. For single erasures, equal-norm Parseval frames are identified as optimal, establishing a unified analysis of spectral geometry and robustness.
Transmitting information reliably across imperfect networks requires mathematical representations that survive data loss. By establishing clear geometric bounds on how Grassmannian frames behave when components are erased, this work helps researchers evaluate and design robust representations for signal transmission and data recovery systems without losing critical structure.
This research is early-stage theoretical mathematics that could inform the design of error-resilient communication protocols, signal recovery tools, and data transmission algorithms. Direct users would primarily be theoretical signal processing researchers and telecommunications system architects. Because the findings are foundational and operator-theoretic, further practical testing and algorithm development are required before real-world commercial deployment can take place.
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Grassmannian frames provide redundant vector representations with low mutual coherence, making them relevant to signal recovery, communication, and erasure-resilient reconstruction. This paper develops an operator-theoretic and geometric framework for analysing finite-dimensional Grassmannian frames through numerical range theory and complex projective geometry. The numerical ranges of the frame operator and Gramian operator are characterised in terms of their spectral extrema. For a frame with eigenvalues bounded by \(\lambda\)min and \(\lambda\)max, the frame operator has numerical range [\(\lambda\)min, \(\lambda\)max], while the Gramian operator has numerical range [0, \(\lambda\)max] when the frame is redundant. Tightness is shown to be equivalent to the degeneration of the frame-operator numerical range to a single point. The Rayleigh map on complex projective space is further related to the numerical range, with its critical points corresponding to eigenvectors of the frame operator. Explicit low-dimensional examples illustrate tight and non-tight configurations and their numerical ranges. Erasure resilience is analysed using masking matrices and associated error operators, leading to lower bounds for the remaining frame operator after multiple erasures. For one erasure, equal-norm Parseval frames are characterised as the optimal Parseval configurations, with worst-case error equal to m/M. The results provide a unified description of spectral geometry, tightness, and erasure robustness for finite-dimensional Grassmannian frame systems.
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DOI: 10.56557/ajpam/2026/v8i1302
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