Remarks on L ¹-solutions of dilation equations

Summary.

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.