Radially symmetric thin plate splines interpolating a circular contour map

Profiles of radially symmetric thin plate spline surfaces minimizing the Beppo Levi energy over a compact annulus $R_{1}\leq r\leq R_{2}$ have been studied by Rabut via reproducing kernel methods. Motivated by our recent construction of Beppo Levi polyspline surfaces, we focus here on minimizing the radial energy over the full semi-axis $0<r<\infty$. Using a $L$-spline approach, we find two types of minimizing profiles: one is the limit of Rabut's solution as $R_{1}\rightarrow0$ and $R_{2}\rightarrow\infty$ (identified as a `non-singular' $L$-spline), the other has a second-derivative singularity and matches an extra data value at $0$. For both profiles and $p\in\left[ 2,\infty\right]$, we establish the $L^{p}$-approximation order $3/2+1/p$ in the radial energy space. We also include numerical examples and obtain a novel representation of the minimizers in terms of dilates of a basis function.Comment: new figures and sub-sections; new Proposition 1 replacing old Corollary 1; shorter proof of Theorem 4; one new referenc

Thin plate splines for transfinite interpolation at concentric circles

We propose a new method for constructing a polyspline on annuli, i.e. a C 2 surface on ℝ2 \ {0}, which is piecewise biharmonic on annuli centered at 0 and interpolates smooth data at all interface circles. A unique surface is obtained by imposing Beppo Levi conditions on the innermost and outermost annuli, and one additional restriction at 0: either prescribing an extra data value, or asking that the surface is non-singular. We show that the resulting Beppo Levi polysplines on annuli are in fact thin plate splines, i.e. they minimize Duchon's bending energy

Transfinite thin plate spline interpolation

Duchon's method of thin plate splines defines a polyharmonic interpolant to scattered data values as the minimizer of a certain integral functional. For transfinite interpolation, i.e. interpolation of continuous data prescribed on curves or hypersurfaces, Kounchev has developed the method of polysplines, which are piecewise polyharmonic functions of fixed smoothness across the given hypersurfaces and satisfy some boundary conditions. Recently, Bejancu has introduced boundary conditions of Beppo Levi type to construct a semi-cardinal model for polyspline interpolation to data on an infinite set of parallel hyperplanes. The present paper proves that, for periodic data on a finite set of parallel hyperplanes, the polyspline interpolant satisfying Beppo Levi boundary conditions is in fact a thin plate spline, i.e. it minimizes a Duchon type functional

The Uniform Convergence of Multivariate Natural Splines

Let f be a function from R to R that has square integrable q-th order partial derivatives, where q ? d=2

Local Accuracy for Radial Basis Function Interpolation on Finite Uniform Grids

We consider interpolation on a finite uniform grid by means of one of the radial basis functions (RBF) OE(r) = r for fl ? 0, fl 62 2N or OE(r) = r ln r for fl 2 2N + . For each positive integer N , set h = N and let fx i : i = 1; 2; : : : ; (N + 1) g be the vertices of the uniform grid of mesh-size h on the unit d-dimensional cube [0; 1] . Given f : [0; 1] ! R, let sh be its unique RBF interpolant at the grid vertices: sh (x i ) = f(x i ), i = 1; 2; : : : ; (N + 1) . For h ! 0, we show that the uniform norm of the error f \Gamma sh on a compact subset K of the interior of [0; 1] enjoys the same rate of convergence to zero as the error of RBF interpolation on the infinite uniform grid hZ , provided that f is a data function whose partial derivatives in the interior of [0; 1] up to a certain order can be extended to Lipschitz functions on [0; 1]