  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 $\left\vert x\right\vert =r_{j}$ for $j=1,...,N$ and satisfy
natural boundary conditions for $\left\vert x\right\vert =r_{1}$ and
$\left\vert x\right\vert =r_{N}.$ 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 \[
\sup_{x\in\Omega}\left\vert f\left( x\right) \right\vert \leq c\sup
_{x\in\Omega}\left\vert \Delta f\left( x\right) \right\vert \] 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.
