Passive Approximation and Optimization Using B-splines

A passive approximation problem is formulated where the target function is an arbitrary complex valued continuous function defined on an approximation domain consisting of a finite union of closed and bounded intervals on the real axis. The norm used is a weighted $\text{L}^p$-norm where $1\leq p\leq\infty$. The approximating functions are Herglotz functions generated by a measure with H\"{o}lder continuous density in an arbitrary neighborhood of the approximation domain. Hence, the imaginary and the real parts of the approximating functions are H\"{o}lder continuous functions given by the density of the measure and its Hilbert transform, respectively. In practice, it is useful to employ finite B-spline expansions to represent the generating measure. The corresponding approximation problem can then be posed as a finite-dimensional convex optimization problem which is amenable for numerical solution. A constructive proof is given here showing that the convex cone of approximating functions generated by finite uniform B-spline expansions of fixed arbitrary order (linear, quadratic, cubic, etc) is dense in the convex cone of Herglotz functions which are locally H\"{o}lder continuous in a neighborhood of the approximation domain, as mentioned above. As an illustration, a typical physical application example is included regarding the passive approximation and optimization of a linear system having metamaterial characteristics.