Two-direction two-scale similarity transformation of two-direction refinable function

In this paper, a special group is used to introduce the two-direction two-scale similarity transform (TDTST) of the two-direction refinable function, and the existence of the solution of the two-direction refinable equation is discussed by the TDTST transform. The approximation order of the two-direction refinable function is discussed by the TDTST transform. Whether the TDTST transform can maintain the orthogonality of the two-direction refinable function is discussed, and it is pointed out that a special real orthogonal matrix can maintain the orthogonality. It is pointed out that the TDTST transform can maintain the symmetry of the two-direction refinable function. Finally, an example is given.