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Multivariate Generalized Hermite Subdivision Schemes

Bin Han

TL;DR

The paper develops a comprehensive multivariate generalization of Hermite subdivision schemes by introducing generalized Hermite schemes of type $\Lambda$ with matrix masks $a$ and a dyadic refinement operator. It connects convergence and smoothness to the sum-rule framework via the smoothness measure $\operatorname{sm}_p(a)$ and the refinable vector function $\phi$, employing a normal-form reduction to analyze vector cascades. A constructive existence theorem guarantees, for any $m$ and $\Lambda\subseteq\Lambda_m$, convergent schemes with spline basis functions and linearly independent integer shifts, including symmetric short-support masks. The work also treats interpolation through generalized Hermite interpolants and linear-phase moments, providing necessary and sufficient conditions and rich multivariate examples, thereby extending univariate Hermite/Lagrange theory to higher dimensions with practical constructions for CAGD and PDE multiscale solvers.

Abstract

Due to properties such as interpolation, smoothness, and spline connections, Hermite subdivision schemes employ fast iterative algorithms for geometrically modeling curves/surfaces in CAGD and for building Hermite wavelets in numerical PDEs. In this paper we introduce a notion of generalized Hermite (dyadic) subdivision schemes and then we characterize their convergence, smoothness and underlying matrix masks with or without interpolation properties. We also introduce the notion of linear-phase moments for achieving the polynomial-interpolation property. For any given positive integer m, we constructively prove that there always exist convergent smooth generalized Hermite subdivision schemes with linear-phase moments such that their basis vector functions are spline functions in $C^m$ and have linearly independent integer shifts. As byproducts, our results resolve convergence, smoothness and existence of Lagrange, Hermite, or Birkhoff subdivision schemes. Even in dimension one our results significantly generalize and extend many known results on extensively studied univariate Hermite subdivision schemes. To illustrate the theoretical results in this paper, we provide examples of convergent generalized Hermite subdivision schemes with symmetric matrix masks having short support and smooth basis vector functions with or without interpolation property.

Multivariate Generalized Hermite Subdivision Schemes

TL;DR

The paper develops a comprehensive multivariate generalization of Hermite subdivision schemes by introducing generalized Hermite schemes of type with matrix masks and a dyadic refinement operator. It connects convergence and smoothness to the sum-rule framework via the smoothness measure and the refinable vector function , employing a normal-form reduction to analyze vector cascades. A constructive existence theorem guarantees, for any and , convergent schemes with spline basis functions and linearly independent integer shifts, including symmetric short-support masks. The work also treats interpolation through generalized Hermite interpolants and linear-phase moments, providing necessary and sufficient conditions and rich multivariate examples, thereby extending univariate Hermite/Lagrange theory to higher dimensions with practical constructions for CAGD and PDE multiscale solvers.

Abstract

Due to properties such as interpolation, smoothness, and spline connections, Hermite subdivision schemes employ fast iterative algorithms for geometrically modeling curves/surfaces in CAGD and for building Hermite wavelets in numerical PDEs. In this paper we introduce a notion of generalized Hermite (dyadic) subdivision schemes and then we characterize their convergence, smoothness and underlying matrix masks with or without interpolation properties. We also introduce the notion of linear-phase moments for achieving the polynomial-interpolation property. For any given positive integer m, we constructively prove that there always exist convergent smooth generalized Hermite subdivision schemes with linear-phase moments such that their basis vector functions are spline functions in and have linearly independent integer shifts. As byproducts, our results resolve convergence, smoothness and existence of Lagrange, Hermite, or Birkhoff subdivision schemes. Even in dimension one our results significantly generalize and extend many known results on extensively studied univariate Hermite subdivision schemes. To illustrate the theoretical results in this paper, we provide examples of convergent generalized Hermite subdivision schemes with symmetric matrix masks having short support and smooth basis vector functions with or without interpolation property.

Paper Structure

This paper contains 15 sections, 12 theorems, 173 equations.

Table of Contents

  1. Introduction and Main Results
  2. Characterize masks of generalized Hermite subdivision schemes
  3. Convergence and smoothness of generalized Hermite subdivision schemes
  4. Characterize refinable generalized Hermite interpolants
  5. Literature review, motivations and contributions
  6. Characterize Masks of Generalized Hermite Subdivision Schemes
  7. Convergence and Smoothness of Generalized Hermite Subdivision Schemes
  8. Normal form of a matrix mask
  9. The smoothness quantity $\operatorname{sm}_p(a)$
  10. Admissible initial vector functions for vector cascade algorithms
  11. Proof of \ref{['thm:converg']}
  12. Generalized Hermite Subdivision Schemes with Interpolation Properties
  13. Existence and Examples of Generalized Hermite Subdivision Schemes
  14. Existence of convergent generalized Hermite subdivision schemes
  15. Examples

Key Result

Theorem 1.1

Let $m\in \mathbb{N}_0, r\in \mathbb{N}$ and an ordered multiset $\Lambda=\{\nu_1,\ldots,\nu_r\} \subseteq \Lambda_m$ as in indset. Suppose that the generalized Hermite subdivision scheme of type $\Lambda$ with a mask $a\in (l_{0}(\mathbb{Z}^d))^{r\times r}$ is convergent with limit functions in $\m If in addition then $\widehat{\phi}(0)\ne 0$ and the following statements hold:

Theorems & Definitions (31)

  • Definition 1
  • Theorem 1.1
  • Theorem 1.2
  • Definition 2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • ...and 21 more