When is a truncated covariance function on the line a covariance function on the circle?

Let [gamma] denote the covariance function of a real stationary process on . Define a new function on [-K, K] by [lambda]K(t) = [gamma](t), t [epsilon] [-K, K]. Note that by identifying the end points of the interval, we may interpret [lambda]K as a function on the circle with circumference 2K. We address the following question: if [gamma] is a covariance function on the line, will [lambda]K be a covariance function on the circle? We identify one class of covariance functions for which the answer is "yes" for all K > 0, and a second class for which it is "yes" for all K sufficiently large. However, our most substantial result is a negative one, and the answer will frequently be "no" for all K > 0. A statistical consequence of the positive results is mentioned briefly.