On cyclic variation-diminishing transforms

We give a new and more manageable characterization for Cyclic Pólya Frequency functions of order 3 (CPF3). Our result also improves present knowledge concerning smoothness properties in CPF. In particular, a conjecture of Mairhuber, Schoenberg, and Williamson, On variation-diminishing transformations on the circle, Rend. Circ. Mat. Palermo (2) 8 (1959), 1-30, about discontinuous CPF functions is established.