The analytic character of real functions f in $C^\infty [a,b]$ which satisfy a certain positivity condition is studied. The condition is of the form $L^k f(x) \geqq 0$, $a \leqq x \leqq b$, $k = 0,1,2, \cdots $ ,where L is a Sturm-Liouville operator and $L^k $ is its kth iterate. It is shown, for a special class of operators, that a function with this property is necessarily the restriction to $[a,b]$ of an analytic function in some complex neighborhood of $[a,b]$. The proof is based on a series representation associated with Sturm-Liouville boundary value problems.