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

Splines Are Universal Solutions of Linear Inverse Problems with Generalized TV Regularization

Splines come in a variety of flavors that can be characterized in terms of some differential operator L. The simplest piecewise-constant model corresponds to the derivative operator. Likewise, one can extend the traditional notion of total variation by considering more general operators than the derivative. This results in the definitions of a generalized total variation seminorm and its corresponding native space, which is further identified as the direct sum of two Banach spaces. We then prove that the minimization of the generalized total variation (gTV), subject to some arbitrary (convex) consistency constraints on the linear measurements of the signal, admits nonuniform L-spline solutions with fewer knots than the number of measurements. This shows that nonuniform splines are universal solutions of continuous-domain linear inverse problems with LASSO, $L _{ 1 }$ , or total-variationlike regularization constraints. Remarkably, the type of spline is fully determined by the choice of L and does not depend on the actual nature of the measurements

Approximation Properties Of Sobolev Splines And The Construction Of Compactly Supported Equivalents

In this paper, we construct compactly supported radial basis functions that satisfy optimal approximation properties. Error estimates are determined by relating these basis functions to the class of Sobolev splines. Furthermore, we derive new rates for approximation by linear combinations of nonuniform translates of the Sobolev splines. Our results extend previous work as we obtain rates for basis functions of noninteger order, and we address approximation with respect to the L-infinity norm. We also use bandlimited approximation to determine rates for target functions with lower order smoothness