Piecewise polynomial functions

For a n-dimensional polyhedral simplex ? embedded in R^nC^r(?) contains C^r piecewise polynomial functions. Thespace C^r_k(?) consisting of all C^r piecewise polynomialfunctions of degree at most k is a finite dimensional vector spaceover R. In this thesis we computed dimensions of C^r_k(?)in the case of n=1 and 2. For n=2 we used two differentmethods. First method is that for a polyhedral complex ? wehave worked on various examples by using CoCoA 4.7 in order tocalculate dimensions of C^r_k(?) when C^r(?) was free.For the second method, we again computed dimensions ofC_k^r(?) for sufficiently large k by using homolojicalcalculations on a simplicial complex ?. We see that on theexamples these two methods coincide.In Chapter 2, we give notations, definitions and results for module.In the first part of Chapter 3, we introduce piecewise polynomialfunctions in one variable. We give necessary and sufficientcondition for the functions to be C^r smooth. Using these fact thedimension formula is given. In the second part, we first studiedpiecewise polynomial function in two variables on a poyhedralcomplex ?, especially on various examples by usingCoCoA 4.7. Secondly, we have studied the relation between idealsof fat points and piecewise polynomial functions on a simplicialcomplex ?. By using this relation together with homolojicalcomputations a formula for dimension of C_k^r(?) insufficiently large degree k is derived.