The Positive-Definite Completion Problem
Abstract
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.
Cite
@article{arxiv.2309.10143,
title = {The Positive-Definite Completion Problem},
author = {Kartik G. Waghmare and Victor M. Panaretos},
journal= {arXiv preprint arXiv:2309.10143},
year = {2023}
}