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Letp,n∈Nwith 2p≥n+ 2,and letIabe a polyharmonic spline oforderpon the gridZ×aZnwhich satisfies the interpolating conditionsIa(j,am) =dj(am) forj∈Z,m∈Znwhere the functionsdj:Rn→Rand the parametera>0 are given. LetBs(Rn) be the set of all integrablefunctionsf:Rn→Csuch that the integralkfks:=ZRnbf(ξ)(1 +|ξ|s)dξis finite. The main result states that for givenσ≥0 there exists aconstantc>0 such that wheneverdj∈B2p(Rn)∩C(Rn),j∈Z,satisfykdjk2p≤D·(1 +|j|σ) for allj∈Zthere exists a polysplineS:Rn+1→Cof orderpon strips such that|S(t,y)−Ia(t,y)|≤a2p−1c·D·(1 +|t|σ)for ally∈Rn,t∈Rand all 0<a≤1

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Convergence of polyharmonic splines on semi-regular grids $\mathbb{Z}{{\boldsymbol{\times}} \boldsymbol{a}} {\mathbb{Z}^{\it\boldsymbol{n}}}$ for ${\boldsymbol{a}\rightarrow\mathbf {0}}$