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

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 $$\varphi(x)=\left(c^\lambda +||x||^\lambda\right)^\beta,\qquad x\in\mathbb{R}^n, \lambda \in\mathbb{R},\beta \in \mathbb{R}\backslash\mathbb{N}, c>0. $$ 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 $\lambda$ and $\beta$ 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.