An extension of a bound for functions in Sobolev spaces, with applications to (m, s)-spline interpolation and smoothing

  • Authors:

  • Rémi Arcangéli;María Cruz López de Silanes;Juan José Torrens

  • Affiliations:

  • Route de Barat, 31160, Arbas, France;Universidad de Zaragoza, Departamento de Matemática Aplicada, C.P.S., María de Luna 3, 50018, Zaragoza, Spain;Universidad Pública de Navarra, Departamento de Ingeniería Matemática e Informática, Campus de Arrosadía, 31006, Pamplona, Spain

  • Venue:
  • Numerische Mathematik
  • Year:
  • 2007

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Abstract

Given a function f on a bounded open subset Ω of $${\mathbb{R}}^n$$ with a Lipschitz-continuous boundary, we obtain a Sobolev bound involving the values of f at finitely many points of $$\overline\Omega$$. This result improves previous ones due to Narcowich et al. (Math Comp 74, 743–763, 2005), and Wendland and Rieger (Numer Math 101, 643–662, 2005). We then apply the Sobolev bound to derive error estimates for interpolating and smoothing (m, s)-splines. In the case of smoothing, noisy data as well as exact data are considered.