Measuring Smoothness of Real-Valued Functions Defined by Sample Points on the Unit Circle

In the context of extracting analytic eigen- or singular values from a polynomial matrix, a suitable cost function is the smoothness of continuous, real, and potentially symmetric periodic functions. This smoothness can be measured as the power of the derivatives of that function, and can be tied to a set of sample points on the unit circle that may be incomplete. We have previously explored the utility of this cost function, and here provide refinements by (i) analysing properties of the cost function and (ii) imposing additional constraints on its evaluation.