Splines minimizing rotation-invariant semi-norms in Sobolev spaces

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


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.


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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  1. [1]
    M. ATTEIA: Etude de certains noyaux et théorie des fonctions "spline" en analyse numérique. Thèse, Grenoble (1966).Google Scholar
  2. [2]
    M. ATTEIA: Existence et détermination des fonctions "spline" à plusieurs variables. C.R. Acad. Sci. Paris 262 (1966), 575–578.MathSciNetzbMATHGoogle Scholar
  3. [3]
    M. ATTEIA: Fonctions "spline" et noyaux reproduisants d'Aronszajn-Bergman. Revue Française d'Informatique et de Recherche Opérationnelle, R-3 (1970), 31–43.MathSciNetGoogle Scholar
  4. [4]
    J. DENY & J.L. LIONS: Les espaces du type de Beppo Levi. Ann. Inst. Fourier 5 (1954), 305–370.MathSciNetCrossRefGoogle Scholar
  5. [5]
    J. DUCHON: Interpolation des fonctions de deux variables suivant le principe de la flexion des plaques minces. Revue Française d'Automatique, Informatique, Recherche Opérationnełle (to appear, 1976).Google Scholar
  6. [6]
    R.L. HARDER & R.N. DESMARAIS: Interpolation using surface splines. J. Aircraft, U.S.A., 9, No 2 (1972), 189–191.CrossRefGoogle Scholar
  7. [7]
    J.L. JOLY: Théorèmes de convergence des fonctions "spline" générales d'interpolation et d'ajustement. C.R. Acad. Sci. Paris 264 (1967),126–128.MathSciNetzbMATHGoogle Scholar
  8. [8]
    P.J. LAURENT: Approximation et Optimisation. Hermann, Paris (1972).zbMATHGoogle Scholar
  9. [9]
    L. SCHWARTZ: Théorie des distributions. Hermann, Paris (1966).zbMATHGoogle Scholar
  10. [10]
    L. SCHWARTZ: Sous-espaces hilbertiens d'espaces vectoriels topologiques et noyaux associés (noyaux reproduisants). J. Anal. Math. 13 (1964), 115–256.zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin · Heidelberg 1977

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques AppliquéesUniversité Scientifique et MédicaleGrenobleFrance

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