Convergence of polyharmonic splines on semi-regular grids ℤ × aℤ^n for a →0

Let p, n ∈ ℕ with 2p ≥ n + 2, and let I a be a polyharmonic spline of order p on the grid ℤ × aℤ n which satisfies the interpolating conditions I_{a}\left( j,am\right) =d_{j}\left( am\right) for j ∈ ℤ, m ∈ ℤ n where the functions d j : ℝ n → ℝ and the parameter a > 0 are given. Let B_{s}\left( \mathbb{R}^{n}\right) be the set of all integrable functions f : ℝ n → ℂ such that the integral \left\| f\right\| _{s}:=\int_{\mathbb{R}^{n}}\left| \widehat{f}\left( \xi\right) \right| \left( 1+\left| \xi\right| ^{s}\right) d\xi 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 C\left( \mathbb{R}^{n}\right) , j ∈ ℤ, satisfy \left\| d_{j}\right\| _{2p}\leq D\cdot\left( 1+\left| j\right| ^{\mathbb{\sigma}}\right) for all j ∈ ℤ there exists a polyspline S : ℝ n+1 → ℂ of order p on strips such that[$] \left| S\left( t,y\right) -I_{a}\left( t,y\right) \right| \leq a^{2p-1}c\cdot D\cdot\left( 1+\left| t\right| ^{\mathbb{\sigma}}\right)[$]for all y ∈ ℝ n , t ∈ ℝ and all 0< a ≤ 1.