دانلود کتاب 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
ight) =d_{j}left( am
ight) 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}
ight) be the set of all integrable functions f : ℝ n → ℂ such that the integral left| f
ight| _{s}:=int_{mathbb{R}^{n}}left| widehat{f}left( xi
ight)
ight| left( 1+left| xi
ight| ^{s}
ight) 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}
ight) cap Cleft( mathbb{R}^{n}
ight) , j ∈ ℤ, satisfy left| d_{j}
ight| _{2p}leq Dcdotleft( 1+left| j
ight| ^{mathbb{sigma}}
ight) for all j ∈ ℤ there exists a polyspline S : ℝ n+1 → ℂ of order p on strips such that[$] left| Sleft( t,y
ight) -I_{a}left( t,y
ight)
ight| leq a^{2p-1}ccdot Dcdotleft( 1+left| t
ight| ^{mathbb{sigma}}
ight)[$]for all y ∈ ℝ n , t ∈ ℝ and all 0< a ≤ 1.