The Correlation Between the Convergence of Subdivision Processes and Solvability of Refinement Equations

Abstract

We consider the univariate two-scale refinement equation phi(chi). This paper analyses the correlation between the existence of smooth compactly supported solutions of this equation and the convergence of the corresponding cascade algorithm/subdivision scheme. In the work 11 we have introduced a criterion that expresses this correlation in terms of the mask of the equation. It is shown that the convergence of subdivision scheme depends on values that the mask takes at the points of its generalized cycles. In this paper we show that the criterion is sharp in the sense that an arbitrary generalized cycle causes the divergence of a suitable subdivision scheme. To do this we construct a general method to produce divergent subdivision schemes having smooth refinable functions. The criterion therefore establishes a complete classification of divergent subdivision schemes.

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Document Details

Document Type
Technical Report
Publication Date
Jul 01, 2001
Accession Number
ADP013750

Entities

People

  • Vladimir Protasov

Organizations

  • Moscow State University

Tags

DTIC Thesaurus Topics

  • Algorithms
  • Coefficients
  • Convergence
  • Eigenvalues
  • Equations
  • Inequalities
  • Mathematics
  • Perturbations
  • Polynomials
  • Sequences
  • Technical Information Centers
  • Trees (Data Structures)

Fields of Study

  • Mathematics

Readers

  • Approximation Theory.
  • Finite Element Method (FEM) for solving Partial Differential Equations (PDEs)