Triangular 3-subdivision schemes: the regular case

The paper deals with the investigation of triangular 3-subdivision schemes in the stationary shift-invariant setting. In Section 2 we collect the available theory on refinable functions (subdivision surfaces), with emphasis on their Sobolev and Hِlder smoothness. Families of interpolatory and approximating 3-subdivision schemes are investigated in Section 3. Some dual 3-subdivision schemes which are related to vector-valued refinable functions are also analyzed. For this purpose, we have developed Matlab routines for numerically investigating properties of vector subdivision schemes.