Multivariate Weak Spline

The multivariate weak spline is defined by piecewise polynomials which smooths only on a set of discrete points. The smooth condition and conformality condition of multivariate weak spline are presented firstly. By these conditions, the dimension and the basis of multivariate weak spline spaces are discussed. In particular, the dimension of multivariate weak spline over the cross-cut partition and triangulation is presented The B net method for studying multivariate weak splines is also discussed. By B net method, the upper bound and low bound of the dimension of multivariate weak splines over triangulation are presented. By the smooth condition of multivariate weak splines, the smooth condition of super splines is presented. According to the smooth condition, the super spline space over type-1 triangulation is discussed. The local supported basis of super spline spaces over type-1 triangulation is presented. By the local supported basis, we build the variation-diminishing operator. The approximation properties of the variation-diminishing operator is also presented.