A Generalization of Schep's Theorem on the Positive Definiteness of a Piecewise Linear Function

Schep proved that, for a piecewise linear function with nodes at integer points, positive definiteness on $\mathbb{R}$ is equivalent to positive definiteness on $\mathbb{Z}$. In this paper, a similar theorem for an entire function of exponential type is proved, and a generalization Schep's theorem is obtained.