On -solutions of a two-direction refinement equation

In this paper we examine L 1 -solutions of the following two-direction refinement equation f ( x ) = ∑ n = − N N c n , 1 f ( k x − n ) + ∑ n = − N N c n , − 1 f ( − k x + n ) . We prove that the vector space of all L 1 -solutions of the above equation is at most one-dimensional and consists of compactly supported functions of constant sign. We also show that in many interesting cases any L 1 -solution of the two-direction refinement equation is either positive or negative on its support. Next we present sufficient conditions (easy for verification) for the existence of nontrivial L 1 -solutions of the two-direction refinement equation as well as for the nonexistence of such solutions.