Hölder Regularity of Geometric Subdivision Schemes

Abstract

We present a framework for analyzing nonlinear \(\mathbb {R}^d\)-valued subdivision schemes which are geometric in the sense that they commute with similarities in \(\mathbb {R}^d\). It allows us to establish \(C^{1,\alpha }\)-regularity for arbitrary schemes of this type, and \(C^{2,\alpha }\)-regularity for an important subset thereof, which includes all real-valued schemes. Safe bounds on the domain of convergence of the scheme and on the Hölder exponent of the limit curves can be found by determining the range of certain real-valued functions. This task can be executed automatically and rigorously by a computer when using interval arithmetics.

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Notes

  1. 1.

    It should be noted that the regularity of the limit curve may indeed depend on the initial data. For instance, this phenomenon can be observed for medianinterpolating subdivision [31], where nonmonotonic data may yield nondifferentiable limits.

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Correspondence to T. Ewald.

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Communicated by Peter Oswald.

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Ewald, T., Reif, U. & Sabin, M. Hölder Regularity of Geometric Subdivision Schemes. Constr Approx 42, 425–458 (2015). https://doi.org/10.1007/s00365-015-9305-3

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Keywords

  • Nonlinear subdivision
  • Geometric subdivision
  • Hölder regularity
  • Convergence
  • Circle-preserving scheme

Mathematics Subject Classification

  • 26A16
  • 68U07