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 φ(x)=(c²ᵈ+||x||²ᵈ)log(c²ᵈ+||x||²ᵈ), xⁿ, d∈ 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)=√c²ᵈ+||x||²ᵈ, xⁿ, d∈ N, c>0, which is restricted to odd dimensions {ortmann}. This generalization improves the approximation order by a factor of O(h²⁽ᵈ⁻¹⁾), 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)=(c^λ +||x||^λ)^β, xⁿ, λ ,β ∈ R, 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≥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 n-1/2 when n is odd. Furthermore, the uniform approximation error is given by O(hⁿ⁻²log(1/h)) for n even and O(hn-3/2) for n odd. Here, $h>0$ denotes the fill distance.