2024/06/23 by Mathis Ortmann, Ortmann, Mathis, Martin Buhmann +1 · 2 citations
Computer Science · Engineering · Mathematics · #Advanced Numerical Analysis Techniques #FOS: Mathematics #Image and Signal Denoising Methods #Iterative Methods for Nonlinear Equations #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2406.16088
openalex publication_date 2024/06/23 · openalex created_date 2024/06/26 · openalex updated_date 2026/07/28
A new generalization of shifted thin plate splines φ(x)=(c2d+||x||2d)log(c2d+||x||2d), x∈ℝn, d∈ ℕ, cgt;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, x∈ℝn, d∈ ℕ, c>0, which is restricted to odd dimensions \citeortmann. 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)=(cλ+||x||λ)β, x∈ℝn, λ∈ℝ,β∈ ℝ\backslashℕ, cgt;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(h(n-3)/(2)) for n odd. Here, h>0 denotes the fill distance.