COMPLEX SPLINES AND AN EXTREMAL PROBLEM

It is well known that there are a lot of papers on the existence of real polynomial spline, but with only a few concerning the complex spline. For example, the existence of such a spline on the unit circle was proved by Ahlberg, Nilson and Walsh in the case of cubic complex spline and non-uniform spacing, and by Schenberg in the case of arbitrary odd degree and equally-spaced knots. In [3], Schmidt proved the existence of complex spline interpolants of arbitrary odd degree with arbitrary spaced knots on the unit circle.