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

Abstract

Let p, n ∈ ℕ with 2p ≥ n + 2, and let I _a be a polyharmonic spline of order p on the grid ℤ × aℤⁿ which satisfies the interpolating conditions $I_{a}\left( j,am\right) =d_{j}\left( am\right)$ for j ∈ ℤ, m ∈ ℤⁿ where the functions d _j : ℝⁿ → ℝ and the parameter a > 0 are given. Let $B_{s}\left( \mathbb{R}^{n}\right)$ be the set of all integrable functions f : ℝⁿ → 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 : ℝⁿ⁺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 ∈ ℝⁿ, t ∈ ℝ and all 0 < a ≤ 1.