Splines and Operators: Part I - Admissibility and Native Spaces

The variational theory of splines requires a native space: a Hilbert space over which the regularization energy $\mathrm{\parallel }\mathrm{L}f{\mathrm{\parallel }}_{L_2}^2$ is well defined and the minimum-energy interpolation problem admits a spline solution. In this paper, we characterize the operators $\mathrm{L}$ that enable the construction of such spaces. Our framework is based on the completion of a space $\mathcal{S}$ of test functions in a controlled Gelfand-triple configuration $\mathcal{S}\subset \mathcal{H}\subset {\mathcal{S}}^{\mathrm{\prime }}$, which we call a Hilbert-Gelfand space. This departs from the classical reproducing-kernel techniques. Our construction is substantially more general, as it supports arbitrary linear operators and singular behaviors, and it recovers the reproducing-kernel setting as a particular case. It is also complete and canonical: the admissibility conditions on $\mathrm{L}$ are necessary and sufficient, and $\mathrm{L}$ determines its native space as the unique Hilbert-Gelfand space on which it extends to a bounded surjection onto $L_2$. We first show that Hilbert-Gelfand spaces are in one-to-one correspondence with compatible Riesz pairs $(\mathrm{A},{\mathrm{A}}^{-1})$ of positive-definite operators $\mathcal{S}\to {\mathcal{S}}^{\mathrm{\prime }}$. We then extend this correspondence to conditional Riesz pairs, where $\mathrm{A}$ is only conditionally positive-definite, which is the scenario that is relevant to splines. The theory identifies the \emph{attainable null space} of the energy as the unique admissible conditioning, realizes the abstract completions as concrete spaces of distributions, and provides a verifiable admissibility condition in terms of a dual energy functional. The framework is illustrated with the fractional Laplacian, which covers polyharmonic and thin-plate splines, and with ordinary differential operators, which cover exponential splines.