In this paper,the dimension of the nonuniform bivariate spline space S_(3)^(1,2)(Δ_(mn)^((2))is discussed based on the theory of multivariate spline space.Moreover,by means of the Conformality of Smoothing Cofactor Method,the basis ofS_(3)^(1,2)(Δ_(mn)^((2))composed of two sets of splines are worked out in the form of the values at ten domain points in each triangular cell,both of which possess distinct local supports.Furthermore,the explicit coefficients in terms of B-net are obtained for the two sets of splines respectively.
Algebraic variety is the most important subject in classical algebraic geometry. As the zero set of multivariate splines, the piecewise algebraic variety is a kind generalization of the classical algebraic variety. This paper studies the correspondence between spline ideals and piecewise algebraic varieties based on the knowledge of algebraic geometry and multivariate splines.
Nther's theorem of algebraic curves plays an important role in classical algebraic geometry. As the zero set of a bivariate spline, the piecewise algebraic curve is a generalization of the classical algebraic curve. Nther-type theorem of piecewise algebraic curves is very important to construct the Lagrange interpolation sets for bivariate spline spaces. In this paper, using the characteristics of quasi-cross-cut partition, properties of bivariate splines and results in algebraic geometry, the Nther-type theorem of piecewise algebraic curves on the quasi-cross-cut is presented.
ZHU ChunGang & WANG RenHong School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
