Error estimates for harmonic and biharmonic interpolation splines with annular geometry

Abstract

The main result in this paper is an error estimate for interpolation biharmonic polysplines in an annulus $A\left(r_1,r_N\right)$, with respect to a partition by concentric annular domains $A\left(r_1,r_2\right),$ ...., $A\left(r_{N-1},r_N\right),$ for radii $0<r_1<....<r_N.$ The biharmonic polysplines interpolate a smooth function on the spheres $\left|x\right|=r_j$ for $j=1,...,N$ and satisfy natural boundary conditions for $\left|x\right|=r_1$ and $\left|x\right|=r_N.$ By analogy with a technique in one-dimensional spline theory established by C. de Boor, we base our proof on error estimates for harmonic interpolation splines with respect to the partition by the annuli $A\left(r_{j-1},r_j\right)$. For these estimates it is important to determine the smallest constant $c\left(\Omega \right),$ where $\Omega =A\left(r_{j-1},r_j\right),$ among all constants $c$ satisfying \[ \sup_{x\inΩ}\left\vert f\left( x\right) \right\vert \leq c\sup _{x\inΩ}\left\vert Δf\left( x\right) \right\vert \] for all $f\in C^2\left(\Omega \right)\cap C\left(\overline{\Omega }\right)$ vanishing on the boundary of the bounded domain $\Omega$ . In this paper we describe $c\left(\Omega \right)$ for an annulus $\Omega =A\left(r,R\right)$ and we will give the estimate \[ \min\{\frac{1}{2d},\frac{1}{8}\}\left( R-r\right) ^{2}\leq c\left( A\left( r,R\right) \right) \leq\max\{\frac{1}{2d},\frac{1}{8}\}\left( R-r\right) ^{2}% \] where $d$ is the dimension of the underlying space.

24 pages