Linear functional equations and Shapiro's conjecture (Q2565939)

Among others the following remarkably general result is offered. Let \(J\subseteq\mathbb{R}\) be a proper interval, \(h\) a complex valued function on \(J,\) \(a_j: J\to\mathbb{C}\) be nowhere vanishing and of bounded variation, \(f_j:\mathbb{R}\to\mathbb{C}\) measurable, \(b_j:J\to\mathbb{R}\) be a difference of two continuous convex functions (\(j=1,\dots ,n\)) and such that \(b_j-b_k\) is not constant on any subinterval of \(J (1\leq j < k\leq n),\) and let \(\sum_{j=1}^n a_j(y)f_j[x+b_j(y)]=h(y)\) be satisfied for almost every \((x,y)\in\mathbb{R}\times J\). Then each \(f_j\) \((j=1,\dots ,n)\) equals almost everywhere an exponential polynomial \(t\mapsto\sum^n_{k=1} \alpha_{jk}e^{\beta_{jk}t}\) \((\alpha_{jk}, \beta_{jk}\in\mathbb{C};\) \(j,k=1,\dots,n).\)