Abstract
Let function $u (z, w) = f (z) h (w)$ be defined on the compact set $\mathbf{K} \subset \mathbb{C}^2$. We study the problem of representation of functions of this class by the product of two continued fractions, which is called a bicontinued fraction. Some properties of Thiele reciprocal derivatives, Thiele continued fractions and regular C-fractions are proved. The possibility of representation of functions of this class by bicontinued fractions is shown. Examples are considered, domains of convergence and uniform convergence of obtained bicontinued fractions to the function are indicated.
Highlights
The functions of one real or complex variable can be approximate polynomials [1, 2, 3], splines [4], rational functions or Pade approximants [5, 6, 7]
We find the coefficients of a continued C–fraction (C–CF) using the formulas (2.18)
If the sequence {ri} is defined as follows r2n−1 = αeαz∗, r2n = e−αz∗, n ∈ N, after equivalent transformations we obtain the expansion of a function eαz into a Thiele continued fraction (T–CF)
Summary
The functions of one real or complex variable can be approximate polynomials [1, 2, 3], splines [4], rational functions or Pade approximants [5, 6, 7]. Continued fractions are used to approximate functions too [8, 9, 10]. The Thiele formula is an analogue of a Taylor formula in the theory of continued fractions. The Thiele formula has advantages over other methods of expansion of a function in continued fractions since coefficients of expansion are determined by reciprocal derivatives of a function [11, 12, 13]. Bicontinued fraction is a new concept that is introduced in this article. The problem of a function approximation by bicontinued fractions investigated in the paper. The new properties of Thiele continued fractions and Thiele reciprocal derivatives are proved. These properties are used when representing functions by bicontinued fractions
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