Multiband electrical filters and Riemann surfaces

We consider another application of Riemann surfaces to the problems of electrical engineering. The synthesis of optimal filters brings us to a certain uniform rational approximation problem which is a multiband extension of 3rd and 4th Zolotarev's problems. Many prominent mathematicians including Zolotarev, Akhiezer, Stiefel and Gonchar were involved in the studies on this topic. With the use of certain algebro-geometric formula (Ansatz) the problem is reduced to the solution of small set of transcendental equations on moduli of Riemann surfaces. The complexity of this approach does not depend on the degree of filter, but rather on the number of its stop- and pass- bands.