2023/09/18 by Kartik G. Waghmare, Waghmare, Kartik G., Victor M. Panaretos +1
Computer Science · Mathematics · #15A83 #46N30 #47A57 #47B32 #Contact Mechanics and Variational Inequalities #FOS: Mathematics #Functional Analysis (math.FA) #Numerical methods in inverse problems #Optimization and Variational Analysis #Probability (math.PR) #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.2309.10143
openalex publication_date 2023/09/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the positive-definite completion problem for kernels on a variety of domains and prove results concerning the existence, uniqueness, and characterization of solutions. In particular, we study a special solution called the canonical completion which is the reproducing kernel analogue of the determinant-maximizing completion known to exist for matrices. We establish several results concerning its existence and uniqueness, which include algebraic and variational characterizations. Notably, we prove the existence of a canonical completion for domains which are equivalent to the band containing the diagonal. This corresponds to the existence of a canonical extension in the context of the classical extension problem of positive-definite functions, which can be understood as the solution to an abstract Cauchy problem in a certain reproducing kernel Hilbert space.