Pictorial technologies for multiband approximation

We consider the best uniform rational approximation of a locally constant two-valued function on several segments of real line. This natural multiband generalization of the fourth Zolotarev problem arises in optimal electrical filter design. Along with complex-analytic and potential theoretic aspects the problem comprises a great deal of combinatorics. To describe the solutions we use the language of colored weighted planar graphs similar to that elaborated for the extremal polynomials. Some modes of the Ansatz method for the 4-band problem will be explained in this language.