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.

翻译：我们研究了多种域上核函数的正定完备化问题，证明了关于解的存在性、唯一性及刻画性质的相关结论。特别地，我们探讨了一种特殊解——典范完备化，该解是矩阵中已知存在的行列式最大化完备化在再生核框架下的对应形式。我们建立了关于其存在性与唯一性的若干结论，包括代数刻画与变分刻画。值得注意的是，我们证明了对于等价于包含对角线的频带域，典范完备化存在。这对应于正定函数经典延拓问题中典范延拓的存在性，可理解为在特定再生核Hilbert空间中抽象柯西问题的解。