Splines minimizing rotation-invariant semi-norms in Sobolev spaces

  • Jean Duchon
Conference paper

DOI: 10.1007/BFb0086566

Part of the Lecture Notes in Mathematics book series (LNM, volume 571)
Cite this paper as:
Duchon J. (1977) Splines minimizing rotation-invariant semi-norms in Sobolev spaces. In: Schempp W., Zeller K. (eds) Constructive Theory of Functions of Several Variables. Lecture Notes in Mathematics, vol 571. Springer, Berlin, Heidelberg

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.

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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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