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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