A class of compactly supported refinable componentwise constant functions in R2

In this paper, we present a class of compactly supported refinable componentwise constant functions with 2×2 dilation matrix A, where A is diagonalizable. A sufficient condition for the compactly supported refinable functions being componentwise constant functions is derived. Furthermore, an iteration algorithm is developed to compute the constant on each component of the functions' support. Finally, two examples are given to illustrate how to use the method proposed to construct the refinable componentwise constant functions. In particular, the second example shows that the componentwise constant function solution of the refinement equation is not locally integrable in R2.