Rational analogues of Bernstein–Szabados operators on several intervals

Bernstein polynomials play a very important role in approximation theory, probability theory, computer aided geometric design and many other areas. In 2017 J. Szabados constructed polynomial operators that can be considered as the most natural generalization to several intervals of the classical Bernstein operators. Their main advantages include fixed difference between degrees of the used polynomials and the number of used nodes. Unfortunately, they exist only under strong restrictions on the geometry of intervals (intervals have to form a polynomial inverse image of an interval). The main goal of the paper is to present a rational operator that generalizes J. Szabados’ construction, and exists for an arbitrary system of several intervals. Moreover, this construction (unlike J. Szabados’) is a linear positive operator. One of the main ingredients in the construction is the fact (which was proved by M.G. Krein, B.Ya. Levin, and A.A. Nudel’man) that an arbitrary finite system of real intervals is the inverse image of an interval by a rational function with precisely one pole at each gap. The approximation properties of such operators are studied as well. Further possible generalizations (of V.S. Videnskii’s operators to one interval) are considered.