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

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Abstract

Let p, n ∈ ℕ with 2pn + 2, and let I a be a polyharmonic spline of order p on the grid ℤ × an 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.

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Kounchev, O., Render, H. Convergence of polyharmonic splines on semi-regular grids \(\mathbb{Z}{{\boldsymbol{\times}} \boldsymbol{a}} {\mathbb{Z}^{\it\boldsymbol{n}}}\) for \({\boldsymbol{a}\rightarrow\mathbf {0}}\) . Numer Algor 44, 255–272 (2007). https://doi.org/10.1007/s11075-007-9099-x

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