Existence results for refinement equations
The authors consider the functional equation \[ f(x)= \sum^N_{n=0} c_nf(kx-n), \quad c_n\geq 0,\;\sum^N_{n=0}c_n=k, \] and seek simple conditions on \(c_n\) under which this equation has nontrivial \(L_1\)-solutions. They introduce the random expression \[ Y=\sum_{n\geq 1} C_nk^{-n} \] where \(C_1, C_2,\dots\) are independent identically distributed random variables distributed as \(C\), where \(P[C=j] =c_j/k\), \(j=0,1, \dots,\mathbb{N}\). The cumulative distribution function \(F\) of \(Y\) satisfies \[ F(x)= {1\over k}\sum^N_{n=0} c_i F(kx-n). \] The authors show that there is a solution to the considered functional equation if and only if the measure whose cumulative distribution function \(F\) is absolutely continuous with respect to Lebesgue measure, and in that case, \(f\) can be taken as the probability density of \(Y\). ``Utilizing this characterization a simple description in terms of the coefficients is given of all refinement equations with no more than six coefficients possessing an integrable solution''.
scientific article; zbMATH DE number 1436162