Remarks on L 1-solutions of dilation equations

Abstract

We give a condition under which the equation $$ f(x) = {\sum\limits_{n = 0}^N {C_{n} f{\left( {\alpha x - \beta _{n} } \right)}} } $$ has no non-trivial L 1-solution. Moreover, we show that the existence of non-trivial L 1-solutions of the dilation equation with given parameters implies the existence of non-trivial L 1-solutions of the dilation equation with other parameters.