论文标题
与环形几何形状的谐波和双旋插插条的误差估计
Error estimates for harmonic and biharmonic interpolation splines with annular geometry
论文作者
论文摘要
本文的主要结果是在Annulus $ a \ left（r_ {1}，r_ {n} \ right）$的插值估算的错误估计中，就同心环形域的分区而言r_ {n-1}，r_ {n} \ right），$ for radii $ 0 <r_ {1} <.... <r_ {n}。$ biharmonic polysplines插值在$ \ left \ left \ welet \ welet \ vert x \ right \ right \ vert x \ pert = r_ witural $ j { x \ right \ vert = r_ {1} $和$ \ weft \ vert x \ right \ right \ right \ vert = r_ {n}。$ by libough，用C. de Boor建立的一维样条理论中的一种技术，我们基于对和谐插图的误差估算的证明，该证明是由Annuli $ a \ awhului $ a \ ewnululi $ a \左（ r_ {j-1}，r_ {j} \ right）$。对于这些估计值，重要的是要确定最小常数$ c \ left（ω\右），$ whene $ω= a \ left（r_ {j-1}，r_ {j} \ right），在所有常数$ c $ cobs usapsying \ [\ sup_ { c\sup _{x\inΩ}\left\vert Δf\left( x\right) \right\vert \] for all $f\in C^{2}\left( Ω\right) \cap C\left( \overline{Ω}\right) $ vanishing on the boundary of the bounded domain $Ω$ .在本文中，我们描述了$ c \ left（ω\右）$的$ω= a \ a \ left（r，r \ right）$，我们将给出估计\ [\ min \ {\ frac {\ frac {1} {1} {2d} {2d} {2d}，\ frac {1} {1} {1} {8} {8} {8} \ lews weft（r左） a \ left（r，r \ right）\ right）\ leq \ max \ {\ frac {1} {2d}，\ frac {1} {8} {8} \} \} \ left（r-r \ r \ rigr）
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 $c\left( Ω\right) ,$ where $Ω=A\left( r_{j-1},r_{j}\right) ,$ 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( Ω\right) \cap C\left( \overline{Ω}\right) $ vanishing on the boundary of the bounded domain $Ω$ . In this paper we describe $c\left( Ω\right) $ for an annulus $Ω=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.
