Related papers: Scalar Subdivision Schemes and Box Splines
In this paper, motivated by applications in computer graphics and animation, we study the numerical methods for checking $C^k-$regularity of vector multivariate subdivision schemes with dilation 2I. These numerical methods arise from the…
Multivariate piecewise polynomial functions (or splines) on polyhedral complexes have been extensively studied over the past decades and find applications in diverse areas of applied mathematics including numerical analysis, approximation…
The paradigm of differentiable programming has significantly enhanced the scope of machine learning via the judicious use of gradient-based optimization. However, standard differentiable programming methods (such as autodiff) typically…
The generation of curves and surfaces from given data is a well-known problem in Computer-Aided Design that can be approached using subdivision schemes. They are powerful tools that allow obtaining new data from the initial one by means of…
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…
We present a completeness characterization of box splines on three-directional triangulations, also called Type-I box spline spaces, based on edge-contact smoothness properties. For any given Type-I box spline, of specific maximum degree…
In this paper we study the dimension of bivariate polynomial splines of mixed smoothness on polygonal meshes. Here, "mixed smoothness" refers to the choice of different orders of smoothness across different edges of the mesh. To study the…
We provide an equivalence between the category of affine, smooth group schemes over the ring of generalized dual numbers $k[I]$, and the category of extensions of the form $1 \rightarrow \text{Lie}(G, I) \rightarrow E \rightarrow G…
We introduce an extension of cone splines and box splines to fractional and complex orders. These new families of multivariate splines are defined in the Fourier domain along certain $s$-dimensional meshes and include as special cases the…
Let $X$ be a smooth quasi-projective algebraic surface and let $\Delta_n$ the big diagonal in the product variety $X^n$. We study cohomological properties of the ideal sheaves $\mathcal{I}^k_{\Delta_n}$ and their invariants…
In this paper we review and refine a technique of Rioul to determine the H\"older regularity of a large class of symmetric subdivision schemes from the spectral radius of a single matrix. These schemes include those of Dubuc and…
Quantum splines are curves in a Hilbert space or, equivalently, in the corresponding Hilbert projective space, which generalize the notion of Riemannian cubic splines to the quantum domain. In this paper, we present a generalization of this…
In a previous paper it was shown that a machine learning regression problem can be solved within the framework of random function theory, with the optimal kernel analytically derived from symmetry and indifference principles and coinciding…
We devise achievable encoding schemes for distributed source compression for computing inner products, symmetric matrix products, and more generally, square matrix products, which are a class of nonlinear transformations. To that end, our…
We provide a new succession rule (i.e. generating tree) associated with Schr\"oder numbers, that interpolates between the known succession rules for Catalan and Baxter numbers. We define Schr\"oder and Baxter generalizations of…
Box splines provide smooth spline spaces as shifts of a single generating function on a lattice and so generalize tensor-product splines. Their elegant theory is laid out in classical papers and a summarizing book. This compendium aims to…
Functions on a bounded domain in scientific computing are often approximated using piecewise polynomial approximations on meshes that adapt to the shape of the geometry. We study the problem of function approximation using splines on a…
Polynomial splines are ubiquitous in the fields of computer aided geometric design and computational analysis. Splines on T-meshes, especially, have the potential to be incredibly versatile since local mesh adaptivity enables efficient…
Normal multi-scale transform [4] is a nonlinear multi-scale transform for representing geometric objects that has been recently investigated [1, 7, 10]. The restrictive role of the exact order of polynomial reproduction $P_e$ of the…
Inclusive deep inelastic scattering factorization combines two features that are often treated separately: an asymptotic reconstruction of the current-current matrix element from hard and long-distance data, and an invariance under finite…