spline wavelets on triangulations

In this paper we investigate spline wavelets on general triangulations. In particular, we are interested in wavelets generated from piecewise quadratic polynomials. By using the Powell-Sabin elements, we set up a nested family of spaces of quadratic splines, which are suitable for multiresolution analysis of Besov spaces. Consequently, we construct wavelet bases on general triangulations and give explicit expressions for the wavelets on the three-direction mesh. A general theory is developed so as to verify the global stability of these wavelets in Besov spaces. The wavelet bases constructed in this paper will be useful for numerical solutions of partial differential equations.