The construction of approximations satisfying the Chebyshev alternance

An efficient algorithm for constructing approximating formulas for sufficiently smoothly changing functions is proposed. Approximating formulas can take the form of a polynomial, a generalized polynomial, a relation of polynomials or generalized polynomials, as well as some function of the listed expressions. The method allows you to find the coefficients of approximating formulas that ensure the achievement of the Chebyshev alternance either for the absolute error or for the relative one. The algorithm is based on an iterative process of finding interpolation nodes. At each iteration, the interpolation nodes are shifted so that the converged process provides the Chebyshev alternance. The method is illustrated on the problem of approximation of the Fermi–Dirac functions, which play an important role in problems of quantum mechanics.