Piecewise Quintic Spline Spaces On Uniform Type-2 Triangulation

Multivariate splines are applied widely in approximation theory, computer aided ge- ometric design and finite element method. In 1975, Ren-Hong Wang established a new approach to study the basic theory on multivariate spline functions by using the meth- ods of function theory and algebraic geometry, and presented the so called Smoothing cofactor-conformality method. Making use of this method, any problem on multivariate spline functions can be studied by transferring it into an equivalent algebraic problem.Multivariate spline function space S_μ^κ(â–³) is a linear space. For a given partitionâ–³,how to find out a set of basic functions that can be easily used in the spline function space S_μ^κ(â–³) is one of the crucial problems. A related problem is to how to determine the dimension of multivariate spline function space S_μ^κ(â–³) denoted by dim S_μ^κ(â–³). [7] and [25] studied lots of spline spaces on type-1 triangulations and type-2 triangulations, which refer to problems of how to determine the dimension of multivariate spline function space, how to find out the basis functions, especially the basis functions with local supports.The uniform type-2 triangulation partition is a special crosscut partition, or a four- directional mesh, which is used widely because of its simple construction and good sym- metry. We discuss the quintic spline spaces on type-2 triangulation partition, which is denoted by S₅³(â–³_mn²) We discuss the dimension of this space. By using the Smoothing cofactor-conformality method, we obtain two B spiles with minimum local supports and spline bases of quintic spline spaces. Furthermore, the corresponding quasi-interpolation operators with high approximation power are constructed.In many practical applications such as building cars, boats, aeroplanes and model- ings, surfaces may not be connected by using the same smoothness degree. Moveover, the need of mathematical research impels us to study the spline functions with different degrees. [2] established splines with different smoothness. [11] discuss the cubic and quartic spline spaces on uniform type-2 triangulation. Considering different smoothness on different grid segments. We study the quintic spline spaces on type-2 triangulation partition with different smoothness on different grid segments, but the space don't have basic spline functions with local supports.