On existence of perfect splines and monosplines with given zeros

Abstract

We provide an elementary proof of existence of periodic function, whose $$$r$$$-th derivative takes only two values, $$$\alpha$$$ and $$$-\beta$$$ ($$$\alpha$$$, $$$\beta$$$ are given positive numbers), and has no more than $$$2n$$$ sign alternations on a period, which has, on a period, exactly $$$2n$$$ zeros, given beforehand. We note that the analogous statement for monosplines easily follows from this.