On Quasi-Interpolation and their associated shift-invariant space using a new class of generalized Thin Plate Splines and Inverse Multiquadrics

TL;DR

The paper advances quasi-interpolation by introducing a generalized, even-dimension-friendly thin plate spline $\varphi(x)=(c^{2d}+||x||^{2d})\log(c^{2d}+||x||^{2d})$ and a broad class of inverse multiquadrics $\varphi(x)=(c^{\lambda}+||x||^{\lambda})^{\beta}$, deriving their Fourier and asymptotic structures. It establishes that, in even dimensions, the quasi-Lagrange operator built from these RBFs reproduces polynomials up to degree $n+2d-1$ and achieves high-order uniform convergence, with an improvement factor of $\mathcal{O}(h^{2(d-1)})$ relative to classical TPS when $d\ge2$, while also detailing how the inverse multiquadrics enable high-order reproduction in higher dimensions via linear combinations. The analysis blends a detailed Fourier-transform decomposition (including Fox $H$-functions and Mellin–Barnes integrals), Strang–Fix-type conditions in Sobolev spaces, and careful asymptotics to characterize viable parameter regimes and reproduction orders. Collectively, the results extend the toolkit for constructing quasi-interpolants with provable, near-best approximation properties across even and odd dimensions, highlighting conditions under which dimensionality strikes can be overcome. Practical impact lies in improved surface reconstruction and data fitting via principled, high-order shift-invariant spaces generated by generalized RBFs.

Abstract

A new generalization of shifted thin plate splines $$\varphi(x)=(c^{2d}+||x||^{2d})\log\left(c^{2d}+||x||^{2d}\right),\qquad x\in\mathbb{R}^n, d\in \mathbb{N}, c>0$$ 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 $\varphi(x)=\sqrt{c^{2d}+||x||^{2d}},\quad x\in\mathbb{R}^n, d\in \mathbb{N}, c>0$, 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 $$\varphi(x)=\left(c^λ+||x||^λ\right)^β,\qquad x\in\mathbb{R}^n, λ\in\mathbb{R},β\in \mathbb{R}\backslash\mathbb{N}, c>0. $$ 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.

Figures (4)

• Figure 1: Visualization of the poles from equation (\ref{['eq: FT IMQ']}) for $\lambda>0, \beta>0$. The singularities of the gamma functions in the nominator are depicted as dots, whereas the singularities of the gamma function in the denominator are represented by circles (or holes), which have the potential to cancel poles. For the purposes of enhanced visualisation, the poles have been represented as if they had an imaginary component, although this is not the case. Every pole lies on the real axis. Additionally, the path of integration is depicted and indicates which poles must be included. The example is provided for $|\lambda|=2, |\beta|=\frac{3}{2}, n=1$.
• Figure 2: Visualization of the poles from equation (\ref{['eq: FT IMQ']}) for $\lambda>0, \beta<0$. The singularities of the gamma functions in the nominator are depicted as dots, while the singularities of the gamma function in the denominator are represented by circles (or holes), which can cancel poles. For purposes of enhanced visualisation, the poles have been represented as if they had an imaginary component, although this is not the case. Every pole lies on the real axis. Additionally, the path of integration is depicted and indicates which poles must be included. The example is provided for $|\lambda|=2, |\beta|=\frac{3}{2}, n=1$.
• Figure 3: Visualization of the poles from equation (\ref{['eq: FT IMQ']}) for $\lambda<0, \beta>0$. The singularities of the gamma functions in the nominator are depicted as dots, while the singularities of the gamma function in the denominator are represented by circles (or holes), which can cancel poles. For purposes of enhanced visualisation, the poles have been represented as if they had an imaginary component, although this is not the case. Every pole lies on the real axis. Additionally, the path of integration is depicted and indicates which poles must be included. The example is provided for $|\lambda|=2, |\beta|=\frac{3}{2}, n=1$.
• Figure 4: Visualization of the poles from equation (\ref{['eq: FT IMQ']}) for $\lambda<0, \beta<0$. The singularities of the gamma functions in the nominator are depicted as dots, while the singularities of the gamma function in the denominator are represented by circles (or holes), which can cancel poles. For purposes of enhanced visualisation, the poles have been represented as if they had an imaginary component, although this is not the case. Every pole lies on the real axis. Additionally, the path of integration is depicted and indicates which poles must be included. The example is provided for $|\lambda|=2, |\beta|=\frac{3}{2}, n=1$.

Theorems & Definitions (9)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8
• Theorem 9