Refinement type equations and Grincevičjus series

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