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 $$\phi (x)=(c^{2d}+\mathrm{\mid }\mathrm{\mid }x\mathrm{\mid }{\mathrm{\mid }}^{2d})\log (c^{2d}+\mathrm{\mid }\mathrm{\mid }x\mathrm{\mid }{\mathrm{\mid }}^{2d}),x\in {\mathbb{R}}^n,d\in \mathbb{N},c>0$$ 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 $\phi (x)=\sqrt{c^{2d}+\mathrm{\mid }\mathrm{\mid }x\mathrm{\mid }{\mathrm{\mid }}^{2d}},x\in {\mathbb{R}}^n,d\in \mathbb{N},c>0$, 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 $$\phi (x)={(c^{\lambda }+\mathrm{\mid }\mathrm{\mid }x\mathrm{\mid }{\mathrm{\mid }}^{\lambda })}^{\beta },x\in {\mathbb{R}}^n,\lambda \in \mathbb{R},\beta \in \mathbb{R}\mathrm{\backslash }\mathbb{N},c>0.$$ 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 $n\ge 3$, 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 $\mathcal{O}(h^{n-2}\log (1\mathrm{/}h))$ for $n$ even and $\mathcal{O}\left(h^{\frac{n-3}{2}}\right)$ for $n$ odd. Here, $h>0$ denotes the fill distance.