On the smoothness of real-valued functions generated by subdivision schemes using nonlinear binary averaging

  • Authors:

  • R. Goldman;E. Vouga;S. Schaefer

  • Affiliations:

  • Department of Computer Science, Rice University, PO Box 1892, MS-132, Houston, TX 77005, USA;Department of Computer Science, Rice University, PO Box 1892, MS-132, Houston, TX 77005, USA;Department of Computer Science, Rice University, PO Box 1892, MS-132, Houston, TX 77005, USA

  • Venue:
  • Computer Aided Geometric Design
  • Year:
  • 2009

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Abstract

Our main result is that two point interpolatory subdivision schemes using C^k nonlinear averaging rules on pairs of real numbers generate real-valued functions that are also C^k. The significance of this result is the following consequence: Suppose that S is a subdivision algorithm operating on sequences of real numbers using linear binary averaging that generates C^m real-valued functions and S@? is the same subdivision procedure where linear binary averaging is replaced everywhere in the algorithm by a C^n nonlinear binary averaging rule on pairs of real numbers; then the functions generated by the nonlinear subdivision scheme S@? are C^k, where k=min(m,n).