A new generalization of shifted thin plate splines \(\begin{aligned} \phi (x)=(c^{2d}+||x||^{2d})\log \left( c^{2d}+||x||^{2d}\right) ,\qquad x\in \mathbb {R}^n, d\in \mathbb {N}, c>0, \end{aligned}\) is presented to further increase the accuracy of quasi-interpolation. When restricted to Euclidean spaces of even dimensionality n, the generalization enables the construction of a quasi-Lagrange operator that reproduces all polynomials of degree \(n+2d-1\) . The case complements the newly proposed generalized multiquadric \(\phi (x)=\sqrt{c^{2d}+||x||^{2d}},\quad x\in \mathbb {R}^n, d\in \mathbb {N}, c>0\) , which is restricted to odd dimensions, as shown in a previous work. This generalization improves the approximation order by a factor of \(\mathcal {O}\left( h^{2(d-1)}\right) \) , where \(h>0\) denotes the fill distance and \(d=1\) represents the classical thin plate spline. The results are then compared with the theoretical optimal approximation from the shift-invariant space generated by this function. Moreover, we introduce a new class of inverse multiquadrics \(\begin{aligned} \phi (x)=\left( c^\lambda +||x||^\lambda \right) ^\beta ,\qquad x\in \mathbb {R}^n, \lambda \in \mathbb {R},\beta \in \mathbb {R}\backslash \mathbb {N}, c>0. \end{aligned}\) 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 \(\mathcal {O}\left( h^{n-2}\log (1/h)\right) \) for n even and \(\mathcal {O}\left( h^{\frac{n-3}{2}}\right) \) for n odd.
