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Splines minimizing rotation-invariant semi-norms in Sobolev spaces

  • Jean Duchon
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 571)

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 ( Open image in new window 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 Open image in new window |t|2m+2s−n (or μ Open image in new window |t|2m+2s−n Log |t|) + polynomials.

Keywords

Sobolev Space Thin Plate Countable Family Surface Spline Closed Linear Subspace 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin · Heidelberg 1977

Authors and Affiliations

  • Jean Duchon
    • 1
  1. 1.Laboratoire de Mathématiques AppliquéesUniversité Scientifique et MédicaleGrenobleFrance

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