Prefractal as the source of new rational approximations of functions with a fractal representation

Abstract

The Article is devoted to the problem of accelerating the convergence of polynomial and rational approximations of functions. In the theory of approximation of functions often used the idea of reducing the interval change in the argument as a method to accelerate the convergence of exponential and rational approximations, approximating this function. In this article, using this idea, a first for this function builds a branching continued fraction, whose branches are either functional series, functional or chain fractions. In this case, the branching continued fraction representing this function is a fractal and at the same time compresses the range of variation of the argument in 2* (k = 1, 2, ...) time, where 2* is the number of branches of the branched chain fraction. That is, the computation of this function at the point x is to compute x/2^k, which is natural and leads to acceleration of convergence of exponential and rational approximations. To build a new rational approximations of branching continued fraction (fractal) is replaced by prefractals - chain-branching fraction with a finite number of links. Here each link is replaced by the polynomial finite chain fraction. In the result, we can obtain arbitrarily many rational approximations.