The cardinal spline interpolant $S_{n,u} $ of degree n to $\exp (iut)$ is shown to satisfy $| {S_{n,u} (t)} | < 1$ unless t is an interpolation point. Also, it is shown that $1 < C_n *#60; 1 + 2^{1 - n} $ for all odd n and $C_n = 1$ for all positive even n, with $C_n: = \sup _{u,t} {{ |\exp (iut) - S_{n,u} (t) |} / {| {u / \pi }} |^{n + 1} }$