Convergence of polyharmonic splines on semi-regular grids mathbb{Z}{{boldsymbol{times}} boldsymbol{a}} {mathbb{Z}^{itboldsymbol{n}}} for {boldsymbol{a}rightarrowmathbf {0}}

Let p, n ∈ ? with 2p ≥ n + 2, and let I _a be a polyharmonic spline of order p on the grid ? × a? ⁿ which satisfies the interpolating conditions $I_{a}left( j,amright) =d_{j}left( amright) $ for j ∈ ?, m ∈ ? ⁿ where the functions d _j : ? ⁿ → ? and the parameter a > 0 are given. Let $B_{s}left( mathbb{R}^{n}right) $ be the set of all integrable functions f : ? ⁿ → ? such that the integral $$ left| fright| _{s}:=int_{mathbb{R}^{n}}left| widehat{f}left( xiright) right| left( 1+left| xiright| ^{s}right) dxi $$ is finite. The main result states that for given $mathbb{sigma}geq0$ there exists a constant c>0 such that whenever $d_{j}in B_{2p}left( mathbb{R}^{n}right) cap Cleft( mathbb{R}^{n}right) ,$ j ∈ ?, satisfy $left| d_{j}right| _{2p}leq Dcdotleft( 1+left| jright| ^{mathbb{sigma}}right) $ for all j ∈ ? there exists a polyspline S : ? ⁿ⁺¹ → ? of order p on strips such that $$ left| Sleft( t,yright) -I_{a}left( t,yright) right| leq a^{2p-1}ccdot Dcdotleft( 1+left| tright| ^{mathbb{sigma}}right) $$ for all y ∈ ? ⁿ , t ∈ ? and all 0 < a ≤ 1.