\(p\)-adic interpolation and approximation of a continuous function by linear combinations of shifts of \(p\)-adic valuations (Q1867264)

In this article the authors consider continuous real functions defined on the ring of \(p\)-adic integers \(Z_p\). Given a set \(x_1, x_2, \ldots, x_{p^n}\) of representatives mod \(p^n\), they consider functions of the form \(x \mapsto \lambda_1 |x - x_1|+ \ldots + \lambda_{p^n} |x - x_{p^n}|\). They show (Theorem 1) that there is a unique such functions with prescribed values at \(x_1, \ldots, x_{p^n}\). Moreover they show that such functions are dense in the space of real continuous functions defined on \(Z_p\) (Theorem 2).