Two-direction poly-scale refinability

The concept of two-direction poly-scale refinable functions with dilation factor aa and shift invariant spaces is introduced. The existence of solutions of two-direction poly-scale refinable equations is investigated. In particular, a necessary and sufficient condition for the orthonormality of the solutions is established. Some properties of two-direction poly-scale refinable functions are discussed. Finally, two examples of two-direction poly-scale refinable functions are given.