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Analysis and convergence of Hermite subdivision schemes

Bin Han

TL;DR

This work analyzes Hermite and vector subdivision schemes with matrix-valued masks, addressing key questions of mask characterization, factorization, and convergence. By studying vector subdivision operators on vector polynomials and connecting Hermite schemes to vector cascade algorithms and refinable vector functions, the authors derive equivalence conditions for polynomial reproduction and reveal the spectral/sum-rule structure underlying convergence. They introduce the normal form of matrix-valued masks to enable mask factorization and reduce analysis to derived schemes, and establish a convergence criterion based on the quantity $\mathrm{sm}_\infty(a)$, showing that convergence to $\mathscr{C}^m$ limiting functions is guaranteed when $\mathrm{sm}_\infty(a)>m$. The results provide practical methods to construct Hermite masks of accuracy order $m+1$ and to verify convergence, with implications for CAGD and Hermite wavelet-based PDE solvers.

Abstract

Hermite interpolation property is desired in applied and computational mathematics. Hermite and vector subdivision schemes are of interest in CAGD for generating subdivision curves and in computational mathematics for building Hermite wavelets to numerically solve partial differential equations. In contrast to well-studied scalar subdivision schemes, Hermite and vector subdivision schemes employ matrix-valued masks and vector input data, which make their analysis much more complicated and difficult than their scalar counterparts. Despite recent progresses on Hermite subdivision schemes, several key questions still remain unsolved, for example, characterization of Hermite masks, factorization of matrix-valued masks, and convergence of Hermite subdivision schemes. In this paper, we shall study Hermite subdivision schemes through investigating vector subdivision operators acting on vector polynomials and establishing the relations among Hermite subdivision schemes, vector cascade algorithms and refinable vector functions. This approach allows us to resolve several key problems on Hermite subdivision schemes including characterization of Hermite masks, factorization of matrix-valued masks, and convergence of Hermite subdivision schemes.

Analysis and convergence of Hermite subdivision schemes

TL;DR

This work analyzes Hermite and vector subdivision schemes with matrix-valued masks, addressing key questions of mask characterization, factorization, and convergence. By studying vector subdivision operators on vector polynomials and connecting Hermite schemes to vector cascade algorithms and refinable vector functions, the authors derive equivalence conditions for polynomial reproduction and reveal the spectral/sum-rule structure underlying convergence. They introduce the normal form of matrix-valued masks to enable mask factorization and reduce analysis to derived schemes, and establish a convergence criterion based on the quantity , showing that convergence to limiting functions is guaranteed when . The results provide practical methods to construct Hermite masks of accuracy order and to verify convergence, with implications for CAGD and Hermite wavelet-based PDE solvers.

Abstract

Hermite interpolation property is desired in applied and computational mathematics. Hermite and vector subdivision schemes are of interest in CAGD for generating subdivision curves and in computational mathematics for building Hermite wavelets to numerically solve partial differential equations. In contrast to well-studied scalar subdivision schemes, Hermite and vector subdivision schemes employ matrix-valued masks and vector input data, which make their analysis much more complicated and difficult than their scalar counterparts. Despite recent progresses on Hermite subdivision schemes, several key questions still remain unsolved, for example, characterization of Hermite masks, factorization of matrix-valued masks, and convergence of Hermite subdivision schemes. In this paper, we shall study Hermite subdivision schemes through investigating vector subdivision operators acting on vector polynomials and establishing the relations among Hermite subdivision schemes, vector cascade algorithms and refinable vector functions. This approach allows us to resolve several key problems on Hermite subdivision schemes including characterization of Hermite masks, factorization of matrix-valued masks, and convergence of Hermite subdivision schemes.

Paper Structure

This paper contains 4 sections, 12 theorems, 123 equations.

Table of Contents

  1. Introduction, Motivations and Main Results
  2. Action of Vector Subdivision Operators on Vector Polynomials
  3. Characterization of Matrix-valued Masks for Hermite Subdivision Schemes
  4. Factorization and Convergence of Hermite Subdivision Schemes

Key Result

Theorem 1.2

Let $a\in (l_{0}(\mathbb{Z}))^{r\times r}$ and $v\in (l_{0}(\mathbb{Z}))^{1\times r}$ with $\widehat{v}(0)\ne 0$. For a nonnegative integer $m\in \mathbb{N}_0$, the following statements are equivalent to each other: Moreover, any of the above items (1)--(6) implies $\mathcal{S}_a(\mathsf{q}*\upsilon_a)=(\mathsf{q}(2^{-1}\cdot))*\upsilon_a$ for all $\mathsf{q}\in \Pi_m$ and

Theorems & Definitions (27)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • ...and 17 more