Powell-Sabin triangulation is obtained by refining a given general triangulation in such a way that each triangle is split into six smaller triangles. The standard construction of continuously differentiable quadratic splines on the refined triangulation is thoroughly researched and due to the known B-spline representation applicable in various fields of numerical analysis. The thesis is concerned with the possibilities of construction of more general polynomial splines on Powell-Sabin triangulations. It contains the analysis of spline spaces of higher degrees and orders of smoothness that extend the space of continuously differentiable quadratic splines. A special attention is paid to the construction of basis B-splines that provide a stable and geometrically intuitive presentation of splines on Powell-Sabin triangulations. The initial part of the thesis is devoted to the overview of elementary results about polynomial splines on triangulations. The quadratic Powell-Sabin splines are presented in more detail. In what follows, cubic splines are considered. The space of continuously differentiable cubic Powell-Sabin splines has not been studied until recently, but it turns out to be very suitable for the use in numerical analysis. Such splines can be represented in terms of B-splines in two different ways. Both representations have a convenient control structure, which is based on a convex partition of unity. One of the representations also allows a simple description of quadratic splines and cubic splines with additional smoothness properties. The last two chapters of the thesis deal with the construction of Powell-Sabin splines of higher degrees. A space of continuously differentiable super splines of degree 4 with many additional smoothness constraints of order two is presented. Furthermore, a family of Powell-Sabin spline spaces with a representative of arbitrary degree is described. The splines of all discussed spaces can be expressed in terms of B-splines that are extensions of quadratic and cubic Powell-Sabin B-splines.
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