In this paper, we study L1 -solutions of the following two-direction poly-scale refinement equation We prove that the vector space of all L1 -solutions of the above equation is at most one-dimensional and consists of compactly supported functions of constant sign. We also show that any- L1 solution of the above equation is either positive or negative on its support under a special assumption. With regard to the - solutions of the equation, some simple sufficient conditions for the existence of nontrivial -solutions and the nonexistence of such solutions are given.
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