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Multivariate Generalized Hermite Subdivision Schemes

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Constructive Approximation Aims and scope

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 integer \(m\in \mathbb {N}\), 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 \(\mathscr {C}^{m}(\mathbb {R}^d)\) and have linearly independent integer shifts. As by-products, 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.

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Authors and Affiliations

  1. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, T6G 2G1, Canada

    Bin Han

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Correspondence to Bin Han.

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Communicated by Peter Oswald

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Research was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC) under Grant RGPIN-2019-04276.

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Han, B. Multivariate Generalized Hermite Subdivision Schemes. Constr Approx 58, 407–462 (2023). https://doi.org/10.1007/s00365-023-09619-3

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  • DOI: https://doi.org/10.1007/s00365-023-09619-3

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