Refinement equations with nonnegative coefficients
In this paper the author analyzes the \(n\)-scale refinement equation \(\Phi(x)= \sum_k p_k \Phi(nx-k)\), where \(n\geq 2\) is an integer, the coefficients \(\{ p_k\}\) are nonnegative, \(\sum_k p_k=1\) and \(\sum_k |k|p_k<\infty\). Under these conditions, a unique solution (up to normalization and addition of a constant) with bounded variation exists, it is real-valued, nondecreasing and either absolutely continuous or purely singular continuous. To decide the continuity type, a tree of order \(n\) is constructed and a sharp criterion involving the zeros of the associated mask \(m_0(\xi)=\sum_k p_k e^{-ik\xi}\) is obtained. The same criterion is applied to the problem of absolute continuity of the distribution function of some random series. Furthermore, the author obtains a complete classification of refinement equations with positive coefficients (in the case of finitely many nonzero \(p_k\)'s) with respect to the existence of continuous or compactly supported solutions.
scientific article; zbMATH DE number 1445626
scientific article