Splines minimizing rotation-invariant semi-norms in Sobolev spaces

Abstract

We define a family of semi-norms ‖μ‖_m,s=(∫_ℝ ⁿ∣τ∣^2s∣ℱ D^mu(τ)∣² d_τ)^1/2 Minimizing such semi-norms, subject to some interpolating conditions, leads to functions of very simple forms, providing interpolation methods that: 1°) preserve polynomials of degree≤m−1; 2°) commute with similarities as well as translations and rotations of ℝⁿ; and 3°) converge in Sobolev spaces H^m+s(Ω).

Typical examples of such splines are: "thin plate" functions ( with Σ λ_a=0, Σ λ_a a=0), "multi-conic" functions (Σ λ_a|t−a|+C with Σ λ_a=0), pseudo-cubic splines (Σ λ_a|t−a|³+α.t+β with Σ λ_a=0, Σ λ_a a=0), as well as usual polynomial splines in one dimension. In general, data functionals are only supposed to be distributions with compact supports, belonging to H^−m−s(ℝⁿ); there may be infinitely many of them. Splines are then expressed as convolutions μ |t|^2m+2s−n (or μ |t|^2m+2s−n Log |t|) + polynomials.