A Deformation Approach to Axially Symmetric Kernels on the Sphere: Harmonic Coupling and Regularity
Abstract
In this paper, we construct axially symmetric kernels on the sphere. We start by taking an isotropic kernel and transform it into an axially symmetric kernel via a specific deformation. For a general deformation, an explicit spherical-harmonic representation is not necessarily available. For our deformation, we can obtain such a representation of the deformed kernel in terms of the expansion coefficients of the original isotropic kernel. We derive this representation by explicitly describing the effect of the deformation on the spherical harmonic basis. From this, an explicit finite sum formula for the coupling coefficients is deduced and we show that, for each fixed input mode, the coefficients decay geometrically at a rate controlled by our deformation parameter. We further note that if the native space of the isotropic kernel is norm equivalent to a Sobolev-space, then the deformed kernel has the same native space with an equivalent norm. In our numerical experiments, we show that our deformation can improve prediction when the underlying random field has an axially symmetric covariance.
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BibTeX
@misc{jager2026deformation,
title = {{A Deformation Approach to Axially Symmetric Kernels on the Sphere: Harmonic Coupling and Regularity}},
author = {Janin Jäger and Fabian Stallbauer},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27419},
eprint = {2609.27419},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27419},
}
RIS
TY - UNPB
TI - A Deformation Approach to Axially Symmetric Kernels on the Sphere: Harmonic Coupling and Regularity
AU - Janin Jäger
AU - Fabian Stallbauer
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.27419
UR - https://arxiv.org/abs/2609.27419
AB - In this paper, we construct axially symmetric kernels on the sphere. We start by taking an isotropic kernel and transform it into an axially symmetric kernel via a specific deformation. For a general deformation, an explicit spherical-harmonic representation is not necessarily available. For our deformation, we can obtain such a representation of the deformed kernel in terms of the expansion coefficients of the original isotropic kernel. We derive this representation by explicitly describing the effect of the deformation on the spherical harmonic basis. From this, an explicit finite sum formula for the coupling coefficients is deduced and we show that, for each fixed input mode, the coefficients decay geometrically at a rate controlled by our deformation parameter. We further note that if the native space of the isotropic kernel is norm equivalent to a Sobolev-space, then the deformed kernel has the same native space with an equivalent norm. In our numerical experiments, we show that our deformation can improve prediction when the underlying random field has an axially symmetric covariance.
ER -