\(L^p\) approximation of completely monotone functions (Q2334383)

A function \(f : [0,\infty) \to [0,\infty)\) is called completely monotone, if \(f\in C[0, \infty) \cap C^\infty(0,\infty)\) and satisfies \((-1)^n f^{(n)}(t)\ge 0\) (\(t>0, \, n=0,1.\dots\)). In the paper under review the authors prove that any completely monotone \(L^p\) function on \([0,\infty)\) is \(\|\cdot\|_p\) limit of a sequence of Dirichlet series with non-negative coefficients. This answers a question of \textit{Y. Liu} [J. Approx. Theory, 112, No. 2, 226--234 (2001; Zbl 0991.30001)].