Error estimates for harmonic and biharmonic interpolation splines with annular geometry
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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( r_1 ,r_2) , ...., A( r_N-1,r_N) , for radii 0<r_1<....<r_N. The biharmonic polysplines interpolate a smooth function on the spheres | x| =r_j for j=1,...,N and satisfy natural boundary conditions for | x| =r_1 and | x| =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( r_j-1,r_j). For these estimates it is important to determine the smallest constant c( Ω) , where Ω=A( r_j-1,r_j) , among all constants c satisfying sup_x∈Ω| f( x) |≤ csup _x∈Ω|Δ f( x) | for all f∈ C^2( Ω) ∩ C( Ω) vanishing on the boundary of the bounded domain Ω . In this paper we describe c( Ω) for an annulus Ω=A( r,R) and we will give the estimate min{1/2d,1/8}( R-r) ^2≤ c( A( r,R) ) ≤max{1/2d,1/8}( R-r) ^2 dimension of the underlying space.
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