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Multivariate Vector Subdivision Schemes with a General Matrix-valued Filter

Ran Lu

TL;DR

This work develops a comprehensive theory for multivariate vector subdivision schemes with general matrix-valued filters by reframing the problem through vector cascade algorithms. It proves that a single, meaningful notion of convergence exists, tying the discrete subdivision process to the continuous cascade dynamics via the $L_p$-smoothness exponent $\operatorname{sm}_p(a,\mathsf{M} I_d)$ and matching filters. The authors provide a rigorous characterization and rate estimates for convergence, establish a mechanism to transform vector schemes into scalar-type forms with strongly invertible filters, and extend results to Lagrange and Hermite-type schemes, backed by explicit 2D examples and construction guidelines. The work advances practical design of smooth refinable vector functions and their applications in CAGD and numerical PDEs by unifying theory, transformation techniques, and concrete constructions. Overall, the framework enables reliable convergence assessment and smoothness control for a broad class of vector subdivision schemes with matrix-valued filters, including balanced and symmetric designs.

Abstract

Subdivision schemes are closely related to splines and wavelets and have numerous applications in CAGD and numerical differential equations. Subdivision schemes employ a scalar filter; that is, scalar subdivision schemes, have been extensively studied in the literature. In contrast, subdivision schemes with a matrix filter, which are the so-called vector subdivision schemes, are far from being well understood. So far, only vector subdivision schemes that use special matrix-valued filters have been well-investigated, such as the Lagrange and Hermite subdivision schemes. To the best of our knowledge, it remains unclear how to define and characterize the convergence of a vector subdivision scheme that uses a general matrix-valued filter. Though filters from Lagrange and Hermite subdivision schemes have nice properties and are widely used in practice, filters not from either subdivision scheme appear in many applications. Hence, it is necessary to study vector subdivision schemes with a general matrix-valued filter. In this paper, from the perspective of a vector cascade algorithm, we show that there is only one meaningful way to define a vector subdivision scheme. We will analyze the convergence of the newly defined vector subdivision scheme and show that it is equivalent to the convergence of the corresponding vector cascade algorithm. Applying our theory, we show that existing results on the convergence of Lagrange and Hermite subdivision schemes can be easily obtained and improved. Finally, we will present some examples of vector subdivision schemes to illustrate our main results.

Multivariate Vector Subdivision Schemes with a General Matrix-valued Filter

TL;DR

This work develops a comprehensive theory for multivariate vector subdivision schemes with general matrix-valued filters by reframing the problem through vector cascade algorithms. It proves that a single, meaningful notion of convergence exists, tying the discrete subdivision process to the continuous cascade dynamics via the -smoothness exponent and matching filters. The authors provide a rigorous characterization and rate estimates for convergence, establish a mechanism to transform vector schemes into scalar-type forms with strongly invertible filters, and extend results to Lagrange and Hermite-type schemes, backed by explicit 2D examples and construction guidelines. The work advances practical design of smooth refinable vector functions and their applications in CAGD and numerical PDEs by unifying theory, transformation techniques, and concrete constructions. Overall, the framework enables reliable convergence assessment and smoothness control for a broad class of vector subdivision schemes with matrix-valued filters, including balanced and symmetric designs.

Abstract

Subdivision schemes are closely related to splines and wavelets and have numerous applications in CAGD and numerical differential equations. Subdivision schemes employ a scalar filter; that is, scalar subdivision schemes, have been extensively studied in the literature. In contrast, subdivision schemes with a matrix filter, which are the so-called vector subdivision schemes, are far from being well understood. So far, only vector subdivision schemes that use special matrix-valued filters have been well-investigated, such as the Lagrange and Hermite subdivision schemes. To the best of our knowledge, it remains unclear how to define and characterize the convergence of a vector subdivision scheme that uses a general matrix-valued filter. Though filters from Lagrange and Hermite subdivision schemes have nice properties and are widely used in practice, filters not from either subdivision scheme appear in many applications. Hence, it is necessary to study vector subdivision schemes with a general matrix-valued filter. In this paper, from the perspective of a vector cascade algorithm, we show that there is only one meaningful way to define a vector subdivision scheme. We will analyze the convergence of the newly defined vector subdivision scheme and show that it is equivalent to the convergence of the corresponding vector cascade algorithm. Applying our theory, we show that existing results on the convergence of Lagrange and Hermite subdivision schemes can be easily obtained and improved. Finally, we will present some examples of vector subdivision schemes to illustrate our main results.
Paper Structure (17 sections, 10 theorems, 116 equations)

This paper contains 17 sections, 10 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Cascade Algorithms and Subdivision Schemes
  3. Convergence of Cascade Algorithms and Subdivision Schemes
  4. Our Contributions and Paper Structure
  5. Convergence of Vector Cascade Algorithms
  6. The Matching Filter $\upsilon_a$ and the Conditions for an Initial Function
  7. Convergence of Vector Cascade Algorithms and the Quantity $\operatorname{sm}_p(a,\mathsf{M} I_d)$
  8. Justification of Definition \ref{['def:vsd']}
  9. Convergence of Vector Subdivision Schemes
  10. Proof of Theorem \ref{['thm:conv:vsd']}
  11. Convergence Rate of a Vector Subdivision Scheme
  12. Transformations of Vector Subdivision Schemes Using Strongly Invertible Filters
  13. Convergence of Lagrange and Hermite Subdivision Schemes
  14. Examples of Vector Subdivision Schemes
  15. Construction Guideline
  16. ...and 2 more sections

Key Result

Theorem 1.2

[hanbook] Let $\mathsf{M}\in\mathbb{N}\setminus\{1\}$ and $m\in\mathbb{N}_0$. Let $a\in l_{0}(\mathbb{Z}^d)$ be such that $\widehat{a}(0)=1$ and $\phi$ be the unique compactly supported distribution that satisfies $\widehat{\phi}(\mathsf{M}\xi)=\widehat{a}(\xi)\widehat{\phi}(\xi)$ and $\widehat{\phi

Theorems & Definitions (24)

  • Definition 1.1
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Proposition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • proof
  • Theorem 3.1
  • proof
  • ...and 14 more