The positive-definite completion problem

Waghmare, Kartik; Panaretos, Victor

doi:10.1090/tran/9194

The positive-definite completion problem
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by Kartik G. Waghmare and Victor M. Panaretos;

Trans. Amer. Math. Soc. 377 (2024), 6549-6594

DOI: https://doi.org/10.1090/tran/9194

Published electronically: June 28, 2024

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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.

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The positive-definite completion problem
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