Technical Report CS0769

TR#:CS0769
Class:CS
Title: OPTIMAL EXTENSIONS OF THE THEOREM OF WORPITZKY FOR CONTINUED FRACTIONS.
Authors: Y. Shapira, A. Sidi and M. Israeli
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Abstract:

One of the best known results in the convergence analysis of continued fractions K(a_n/1) is the theorem of Worpitzky. This theorem states that for |a_n| \leq 1/4, n = 1,2,..., convergence takes place and the approximants w_n, n = 1,2,..., and the limits w of the continued fractions are in the circular region |z| \leq 1/2. In this work we prove new results under the condition |a_n| \leq r/4,\ n = 1,2,...,\ 0 < r \leq 1, that are also optimal. In particular, we show that the approximants and the limits are in the circular region |z| \leq (1-\sqrt{1-r})/2, and that the errors w_n - w satisfy |w_n-w| \leq |v_n-v|, where v_n, n = 1,2,..., and v are the approximants and limit, respectively, of the continued fraction K(a_n/1) for which a_n = - r/4, n = 1,2,...\ . The theorem of Worpitzky follows from this result by letting r = 1. Another consequence is that, for 0 < r < 1, W_N - W = O([(1-\Sqrt{1-R})/(1+\Sqrt{1-R})]^N) at least, as n \rightarrow \infty.

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