Splinets 1.5.0 -- Periodic Splinets

Splinets -- efficient orthonormalization of the B-splines

A new efficient orthogonalization of the B-spline basis is proposed and contrasted with some previous orthogonalized methods. The resulting orthogonal basis of splines is best visualized as a net of functions rather than a sequence of them. For this reason, the basis is referred to as a splinet. The splinets feature clear advantages over other spline bases. They efficiently exploit 'near-orthogonalization' featured by the B-splines and gains are achieved at two levels: locality that is exhibited through small size of the total support of a splinet and computational efficiency that follows from…

Splinets——通过泰勒展开构造的样条、其支撑集与正交基

Splinets -- splines through the Taylor expansion, their support sets and orthogonal bases

本文实现了基于splinets正交基和支撑集的样条新表示方法，利用泰勒展开和二进正交化算法，为函数数据分析提供了高效且稳定的计算工具，尤其适用于稀疏数据场景。

用于密度函数表示的高效样条正交基

Efficient spline orthogonal basis for representation of density functions

本文针对密度函数样条表示中标准B-样条不满足零积分约束且现有ZB-样条缺乏正交性的问题，提出构造正交ZB-样条基的高效方法ZB-splinets，兼具计算高效性与局部支撑优势，并在人口统计数据上验证。

Multiply Periodic Splines

Spline functions have long been used in numerical solution of differential equations. Recently it revives as isogeometric analysis, which offers integration of finite element analysis and NURBS based CAD into a single unified process. Usually many NURBS pieces are needed to build geometrically continuous CAD models. In this paper, we introduce some multiply periodic splines defined on hyperbolic disc. A single piece of such splines is enough to build complex CAD models. Multiresolution analysis on surfaces of high genus built from such splines can be carried out naturally. CAD and FEA are inte…

On the stability of the $L^{2}$ projection and the quasiinterpolant in the space of smooth periodic splines

In this paper we derive stability estimates in $L^{2}$- and $L^{\infty}$- based Sobolev spaces for the $L^{2}$ projection and a family of quasiinterolants in the space of smooth, 1-periodic, polynomial splines defined on a uniform mesh in $[0,1]$. As a result of the assumed periodicity and the uniform mesh, cyclic matrix techniques and suitable decay estimates of the elements of the inverse of a Gram matrix associated with the standard basis of the space of splines, are used to establish the stability results.

基于B-样条拟插值的Smolyak网格上的高维周期采样

High-dimensional periodic sampling on Smolyak grids based on B-spline quasi-interpolation

本文基于B-样条拟插值，构建了在Smolyak网格上具有Lipschitz-Hölder混合光滑度α>0的高维周期函数的采样恢复和求积公式线性算法，并研究了其最优性。