Rational Interpolation: Analytical Solution

Abstract

The classical problem of interpolation by rational functions is well known to reduce to a system of linear algebraic equations, but the resulting system is usually complicated for qualitative analysis and numerical implementation. We propose a new approach which generalizes the polynomial interpolation to the rational interpolation both in the statement of the problem and its study. We present explicit formulas for solution, as well as give simple sufficient conditions for existence of a solution and describe the set of unsolvable problems. Also, we provide a basis for effective numerical implementation. According to the modern terminology of function theory, we study multipoint Pade approximations.