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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2406.16088 |
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| _version_ | 1866910499519594496 |
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| author | Ortmann, Mathis Buhmann, Martin |
| author_facet | Ortmann, Mathis Buhmann, Martin |
| contents | A new generalization of shifted thin plate splines
$$φ(x)=(c^{2d}+||x||^{2d})\log\left(c^{2d}+||x||^{2d}\right),\qquad x\in\mathbb{R}^n, d\in \mathbb{N}, c>0$$ is presented to increase the accuracy of quasi-interpolation further. With the restriction to Euclidean spaces of even dimensionality, the generalization can be used to generate a quasi-Lagrange operator that reproduces all polynomials of degree $n+2d-1$. It thus complements the case of the newly proposed generalized multiquadric $φ(x)=\sqrt{c^{2d}+||x||^{2d}},\quad x\in\mathbb{R}^n, d\in \mathbb{N}, c>0$, which is restricted to odd dimensions \cite{ortmann}. This generalization improves the approximation order by a factor of $\mathcal{O}\left(h^{2(d-1)}\right)$, where $d=1$ represents the classical thin plate spline. The results are then compared with the theoretical optimal approximation from the shift-invariant space generated by these functions. Moreover, we introduce a new class of inverse multiquadrics $$φ(x)=\left(c^λ+||x||^λ\right)^β,\qquad x\in\mathbb{R}^n, λ\in\mathbb{R},β\in \mathbb{R}\backslash\mathbb{N}, c>0. $$ We provide an explicit representation of the generalized Fourier transform and discuss its asymptotic behaviour near the origin. Particular emphasis is placed on the case where $λ$ and $β$ are both negative. It is demonstrated that, in dimensions $n\geq3$, it is possible to build a quasi-Lagrange operator that reproduces all polynomials of degree $n-3$ when $n$ is even and of degree $\frac{n-1}{2}$ when n is odd. Furthermore, the uniform approximation error is given by $\mathcal{O}\left(h^{n-2}\log(1/h)\right)$ for $n$ even and $\mathcal{O}\left(h^{\frac{n-3}{2}}\right)$ for $n$ odd. Here, $h>0$ denotes the fill distance. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_16088 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Quasi-Interpolation and their associated shift-invariant space using a new class of generalized Thin Plate Splines and Inverse Multiquadrics Ortmann, Mathis Buhmann, Martin Numerical Analysis A new generalization of shifted thin plate splines $$φ(x)=(c^{2d}+||x||^{2d})\log\left(c^{2d}+||x||^{2d}\right),\qquad x\in\mathbb{R}^n, d\in \mathbb{N}, c>0$$ is presented to increase the accuracy of quasi-interpolation further. With the restriction to Euclidean spaces of even dimensionality, the generalization can be used to generate a quasi-Lagrange operator that reproduces all polynomials of degree $n+2d-1$. It thus complements the case of the newly proposed generalized multiquadric $φ(x)=\sqrt{c^{2d}+||x||^{2d}},\quad x\in\mathbb{R}^n, d\in \mathbb{N}, c>0$, which is restricted to odd dimensions \cite{ortmann}. This generalization improves the approximation order by a factor of $\mathcal{O}\left(h^{2(d-1)}\right)$, where $d=1$ represents the classical thin plate spline. The results are then compared with the theoretical optimal approximation from the shift-invariant space generated by these functions. Moreover, we introduce a new class of inverse multiquadrics $$φ(x)=\left(c^λ+||x||^λ\right)^β,\qquad x\in\mathbb{R}^n, λ\in\mathbb{R},β\in \mathbb{R}\backslash\mathbb{N}, c>0. $$ We provide an explicit representation of the generalized Fourier transform and discuss its asymptotic behaviour near the origin. Particular emphasis is placed on the case where $λ$ and $β$ are both negative. It is demonstrated that, in dimensions $n\geq3$, it is possible to build a quasi-Lagrange operator that reproduces all polynomials of degree $n-3$ when $n$ is even and of degree $\frac{n-1}{2}$ when n is odd. Furthermore, the uniform approximation error is given by $\mathcal{O}\left(h^{n-2}\log(1/h)\right)$ for $n$ even and $\mathcal{O}\left(h^{\frac{n-3}{2}}\right)$ for $n$ odd. Here, $h>0$ denotes the fill distance. |
| title | On Quasi-Interpolation and their associated shift-invariant space using a new class of generalized Thin Plate Splines and Inverse Multiquadrics |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2406.16088 |