Multivariate Generalized Hermite Subdivision Schemes

Hermite interpolation and more generally Birkhoff interpolation are useful in mathematics and applied sciences. Due to their many desired properties such as interpolation, smoothness, short support and spline connections of basis functions, multivariate Hermite subdivision schemes employ fast numerical algorithms for geometrically modeling curves/surfaces in CAGD and isogeometric analysis, and for building Hermite wavelet methods in numerical PDEs and data sciences. In contrast to recent extensive study of univariate Hermite subdivision schemes, multivariate Hermite subdivision schemes are barely analyzed yet in the literature. In this paper we first introduce a notion of generalized Hermite subdivision schemes including all Hermite and other subdivision schemes as special cases. Then we analyze and characterize generalized Hermite masks, convergence and smoothness of generalized Hermite subdivision schemes with or without interpolation properties. We also introduce the notion of linear-phase moments for generalized Hermite subdivision schemes to have the polynomial-interpolation property. We constructively prove that there exist convergent smooth generalized Hermite subdivision schemes (including Hermite, Lagrange or Birkhoff subdivision schemes) with linear-phase moments such that their basis vector functions are spline functions in $C^m$ for any given integer $m$ and have linearly independent integer shifts. Our results not only extend and analyze many variants of Hermite subdivision schemes but also resolve and characterize as byproducts convergence and smoothness of Hermite, Lagrange or Birkhoff subdivision schemes. Even in dimension one our results significantly generalize and extend many known results on univariate Hermite subdivision schemes. Examples are provided to illustrate the theoretical results in this paper.