Date: 02 Oct 2006

Splines minimizing rotation-invariant semi-norms in Sobolev spaces

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Abstract

We define a family of semi-norms ‖μ‖m,s=(∫ n∣τ∣2s∣ℱ Dmu(τ)∣2 dτ)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 Hm+s(Ω).

Typical examples of such splines are: "thin plate" functions ( with Σ λa=0, Σ λa a=0), "multi-conic" functions (Σ λa|t−a|+C with Σ λa=0), pseudo-cubic splines (Σ λa|t−a|3+α.t+β with Σ λa=0, Σ λa 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.