FUNCTIONAL-EQUATION

We consider the functional equation f(i)(x) = SIGMA(j) a(ij)f(tau(j)x) + h(i)(x) where a(ij) and tau(j) are fixed constants, 0 < tau(j) < 1, and h(i) is a given function. The independant real variable x runs either on ]0, x(0)[ or on ]0, +infinity[. We give necessary and sufficient conditions of algebraic type in order that the linear mapping h bar arrow pointing right f be well-defined and continuous from a Sobolev-type space (one may think to L2(dx/x)) into itself. A former analysis within the context of differentiable functions is due to Le Floch and Li Ta Tsien [1].