Integral Representations Of Positive Definite Kernels For Second-Order Elliptic Operators
DOI:
https://doi.org/10.31471/2304-7399-2026-22(83)-18-28Keywords:
positive definite kernel, integral representations, elliptic operator, spectral measureAbstract
The paper investigates positive definite kernels of two variables associated with an elliptic differential expression of the second order containing the lower coefficients. For the specified class of operators, a necessary and sufficient condition for the existence of an integral image of the kernel in the form of an expansion in the fundamental system of operator-generated functionals with respect to a matrix-valued analytic measure is established.
The proposed construction generalizes the classical scheme of integral images of the Laplace type to the case of non-self-adjoint differential structures. The obtained results provide a constructive parameterization of the class of admissible kernels through the spectral data of the corresponding operator, which opens up new possibilities for the analysis of correlation functions in the theory of random fields and spectral theory.
References
1. V. I. Paulsen, M. Raghupathi, An Introduction to the Theory of Reproducing Kernel Hilbert Spaces, Cambridge University Press, 2016. https://doi.org/10.1017/CBO9781316219232
2. S. Saitoh, Y. Sawano, Theory of Reproducing Kernels and Applications, Springer, 2016. https://doi.org/10.1007/978-981-10-0530-5
3. G. E. Fasshauer, M. J. McCourt, Kernel-based Approximation Methods Using MATLAB, World Scientific, 2015. doi:10.1142/9789814630153
4. V. Borovitskiy, A. Terenin, P. Mostowsky, M. P. Deisenroth, Matérn Gaussian Processes on Riemannian Manifolds, in: Advances in Neural Information Processing Systems 33 (NeurIPS 2020), 2020. https://doi.org/10.48550/arXiv.2006.10160
5. M. Hutchinson, A. Terenin, V. Borovitskiy, S. Takao, Y. W. Teh, M. P. Deisenroth, Vector-valued Gaussian Processes on Riemannian Manifolds via Gauge Independent Projected Kernels, in: Advances in Neural Information Processing Systems 34 (NeurIPS 2021), 2021. https://doi.org/10.48550/arXiv.2110.14423
6. Ju. M. Berezanskii, Expansions in eigenfunctions of selfadjoint operators, Translations of Mathematical Monographs, Vol. 17, American Mathematical Society, Providence, R.I., 1968, 809 p. https://bookstore.ams.org/mmono-17
7. P. Batlle, M. D. Darcy, B. Hosseini, H. Owhadi, Kernel methods are competitive for operator learning, Journal of Computational Physics 496 (2024), 112549. https://doi.org/10.1016/j.jcp.2023.112549
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Copyright (c) 2026 Ivanna Andrusiak, Оксана Бродяк
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