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 (x) = (c^2d+||x||^2d) (c^2d+||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 (x) =c^2d+||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 (h^2 (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 (x) = (c^ +||x||^) ^, xⁿ, , 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 and are both negative. It is demonstrated that, in dimensions n3, 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-12 when n is odd. Furthermore, the uniform approximation error is given by O (h^n-2 (1/h) ) for n even and O (h^n-3{2}) for n odd. Here, h>0 denotes the fill distance.