On L1L1-solutions of a two-direction refinement equation

In this paper we examine L1L1-solutions of the following two-direction refinement equationf(x)=∑n=−NNcn,1f(kx−n)+∑n=−NNcn,−1f(−kx+n). We prove that the vector space of all L1L1-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 L1L1-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 L1L1-solutions of the two-direction refinement equation as well as for the nonexistence of such solutions.