Vector field multiplier operators and matrix-valued kernel quasi-interpolation
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| Format: | Preprint |
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2026
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| _version_ | 1866918487007428608 |
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| author | Sun, Zhengjie Huang, Biao Sun, Xingping |
| author_facet | Sun, Zhengjie Huang, Biao Sun, Xingping |
| contents | We develop and analyze a class of matrix-valued spherical-convolution kernels stemming from scaled zonal functions on $\mathbb{S}^2,$ the unit sphere embedded in $\mathbb{R}^3$. The construct of these kernels utilizes the Legendre differential equation and requires less stringent regularity conditions on the original zonal kernels. The induced integral operators are simple Fourier-Legendre multipliers that not only deliver optimal Sobolev error estimates (in terms of the scaling parameter) but also yield natural Helmholtz-Hodge decompositions on the $L_2$-tangential vector fields on $\mathbb{S}^2$. Via discretization of the underlying convolution integrals, we harvest a family of vector-valued quasi-interpolants that accomplish our approximation goal in the divergence/curl-free vector field. The quasi-interpolation algorithm is robust against noisy data. The implementation process is adaptive to human-improvision, involving neither evaluating the convolution integrals nor solving systems of linear equations. The computational efficiency and executory robustness of the quasi-interpolation algorithm stand in sharp contrast to the existing kernel-based vector field interpolation method. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_05610 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Vector field multiplier operators and matrix-valued kernel quasi-interpolation Sun, Zhengjie Huang, Biao Sun, Xingping Numerical Analysis 43A90, 41A25, 41A55, 65D12, 65D32 We develop and analyze a class of matrix-valued spherical-convolution kernels stemming from scaled zonal functions on $\mathbb{S}^2,$ the unit sphere embedded in $\mathbb{R}^3$. The construct of these kernels utilizes the Legendre differential equation and requires less stringent regularity conditions on the original zonal kernels. The induced integral operators are simple Fourier-Legendre multipliers that not only deliver optimal Sobolev error estimates (in terms of the scaling parameter) but also yield natural Helmholtz-Hodge decompositions on the $L_2$-tangential vector fields on $\mathbb{S}^2$. Via discretization of the underlying convolution integrals, we harvest a family of vector-valued quasi-interpolants that accomplish our approximation goal in the divergence/curl-free vector field. The quasi-interpolation algorithm is robust against noisy data. The implementation process is adaptive to human-improvision, involving neither evaluating the convolution integrals nor solving systems of linear equations. The computational efficiency and executory robustness of the quasi-interpolation algorithm stand in sharp contrast to the existing kernel-based vector field interpolation method. |
| title | Vector field multiplier operators and matrix-valued kernel quasi-interpolation |
| topic | Numerical Analysis 43A90, 41A25, 41A55, 65D12, 65D32 |
| url | https://arxiv.org/abs/2605.05610 |