Stability of refinable functions, multiresolution analysis, and Haar bases

The stability of the integer translates of a univariate refinable function is characterized in terms of the mask sequence in the corresponding k-scale (k greater than or equal to 2) refinement equation. We show that the stability and refinement of some kinds of basis functions lead to a multiresolution analysis in L(p)(R(s))(1 less than or equal to p less than or equal to infinity, s is an element of N) based on general lattices. As an application we determine explicitly all those multiresolution analyses in L(2)(R) associated with (Z, k) whose scaling functions are characteristic functions.