Refinement type equations and Grincevičjus series

We consider L1L1-solutions of the following refinement type equationsf(x)=∑n∈Zcn,1f(kx−n)+∑n∈Zcn,−1f(−kx−n), where k⩾2k⩾2 is an integer and for all n∈Zn∈Z reals cn,1cn,1, cn,−1cn,−1 are non-negative with ∑n∈Z(cn,1+cn,−1)=k∑n∈Z(cn,1+cn,−1)=k and ∑n∈Zlog|n|(cn,1+cn,−1)<+∞∑n∈Zlog|n|(cn,1+cn,−1)<+∞. Necessary and sufficient conditions for the existence of non-trivial L1L1-solutions in several special cases are given.