Error estimates for harmonic and biharmonic interpolation splines with annular geometry

The main result in this paper is an error estimate for interpolation biharmonic polysplines in an annulus $A(r_1,r_N)$, with respect to a partition by concentric annular domains $A\left( r_{1} ,r_{2}\right) ,$\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.},$A\left( r_{N-1},r_{N}\right) ,$forradii$0<r_{1}<…<r_{N}.$ The biharmonic polysplines interpolate a smooth function on the spheres $\mid x\mid =r_j$ for $j=1,\mathrm{.}\mathrm{.}\mathrm{.},N$ and satisfy natural boundary conditions for $\mid x\mid =r_1$ and $\mid x\mid =r_N\mathrm{.}$ 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 $A(r_{j-1},r_j)$. For these estimates it is important to determine the smallest constant $c\left(\mathrm{\Omega }\right),$ where $\mathrm{\Omega }=A(r_{j-1},r_j),$ among all constants $c$ satisfying $$\underset{x\in \mathrm{\Omega }}{\sup }\mid f\left(x\right)\mid \le c\underset{x\in \mathrm{\Omega }}{\sup }\mid \mathrm{\Delta }f\left(x\right)\mid$$ for all $f\in C^{2}\left( \Omega\right) \cap C\left( \overline{\Omega }\right) $ vanishing on the boundary of the bounded domain $\mathrm{\Omega }$ . In this paper we describe $c\left(\mathrm{\Omega }\right)$ for an annulus $\mathrm{\Omega }=A(r,R)$ and we will give the estimate $$\min \{\frac{1}{2d},\frac{1}{8}\}{(R-r)}^2\le c\left(A(r,R)\right)\le \max \{\frac{1}{2d},\frac{1}{8}\}{(R-r)}^2$$ where $d$ is the dimension of the underlying space.