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Journals / Constructive Mathematical Analysis / 2026 / Volume: 9 Issue: 3

Finite-rank kernel realization and completion

James Tian(Mathematical Reviews)0000-0003-4996-1227
Pages
—
Publication Date
2026-09-15
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

We also give a rank bound for finite completions obtained by joining two completed pieces along their common feature subspace.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:We consider when local finite-rank positive definite kernels come from a single global finite-rank kernel.
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Key Findings & Data:In that setting, rank-preserving completion becomes a unitary patching problem on overlap generated subspaces.
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Conclusion & Impact:We also give a rank bound for finite completions obtained by joining two completed pieces along their common feature subspace.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Positive definite kernels · finite-rank kernels · Kolmogorov factorization

Abstract

We consider when local finite-rank positive definite kernels come from a single global finite-rank kernel. The first part treats the case where the global kernel is fixed and shows that the missing mixed blocks control when a global rank-$r$ realization exists and how rank grows when it does not. The second part treats the completion problem, where only local kernels are given. In that setting, rank-preserving completion becomes a unitary patching problem on overlap generated subspaces. Forests always patch, full overlaps are governed by cycle conditions for the induced unitaries, and partial overlaps lead to finite-dimensional subspace compatibility conditions. We also give a rank bound for finite completions obtained by joining two completed pieces along their common feature subspace.

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Cite This Article

Tian, J. (2026). Finite-rank kernel realization and completion. Constructive Mathematical Analysis, Volume: 9 Issue: 3. https://doi.org/10.33205/cma.1942507

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