Splines and Operators: Part I - Admissibility and Native Spaces

Abstract

The variational theory of splines requires a native space: a Hilbert space over which the regularization energy |L f|_L₂² is well defined and the minimum-energy interpolation problem admits a spline solution. In this paper, we characterize the operators L that enable the construction of such spaces. Our framework is based on the completion of a space 𝒮 of test functions in a controlled Gelfand-triple configuration 𝒮⊂ℋ⊂𝒮', 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 L are necessary and sufficient, and L determines its native space as the unique Hilbert-Gelfand space on which it extends to a bounded surjection onto L₂. We first show that Hilbert-Gelfand spaces are in one-to-one correspondence with compatible Riesz pairs (A,A⁻¹) of positive-definite operators 𝒮→𝒮'. We then extend this correspondence to conditional Riesz pairs, where 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.

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BibTeX RIS

BibTeX

@misc{unser2026splines,
  title = {{Splines and Operators: Part I - Admissibility and Native Spaces}},
  author = {Michael Unser},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.14906},
  eprint = {2610.14906},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.14906},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Splines and Operators: Part I - Admissibility and Native Spaces
AU  - Michael Unser
PY  - 2026
DA  - 2026/10/10
PB  - arXiv
DO  - 10.48550/arxiv.2610.14906
UR  - https://arxiv.org/abs/2610.14906
AB  - The variational theory of splines requires a native space: a Hilbert space over which the regularization energy |L f|_L_2^2 is well defined and the minimum-energy interpolation problem admits a spline solution. In this paper, we characterize the operators L that enable the construction of such spaces. Our framework is based on the completion of a space 𝒮 of test functions in a controlled Gelfand-triple configuration 𝒮⊂ℋ⊂𝒮', 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 L are necessary and sufficient, and 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 (A,A^−1) of positive-definite operators 𝒮→𝒮'. We then extend this correspondence to conditional Riesz pairs, where 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.
ER  -

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