In this chapter we shall consider rational approximation of functions with respect to the Hausdorff distance. The Hausdorff distance in the space C[a, b] of the continuous functions in the interval [a, b] was introduced by Bl. Sendov and B. Penkov (1962). After this Bl. Sendov developed the theory of approximation of bounded functions by means of algebraic polynomials with respect to the Hausdorff distance. Many mathematicians have obtained results in the theory of approximation of functions with respect to the Hausdorff distance – the results are collected in the book of Bl. Sendov (1979).

In section 9.1 we give the definition of Hausdorff distance in the set of all bounded functions in a given interval and we consider some of its properties.

In section 9.2 we consider the most interesting examples of rational approximation in Hausdorff distance – rational approximation of sign x. In our opinion this result is basic in the theory of rational approximation – from here follows the most essential results for uniform and Lp rational approximation – for example Newman's result for |x|. The Hausdorff distance is the natural distance by means of which we can explain the fact that sign x can be approximated to order O(e−c√n) by means of rational functions.