Über Folgen,die BezÜglich Eines Mittels Rekurrent Sind |
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Authors: | Peter Flor Franz Halter-Koch |
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Institution: | 1. Institut für Mathematik, Karl-Franzens-Universit?t, Heinrichstrasse 36/IV, A-8010, Graz, ?sterreich
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Abstract: | Let X be a convex subset of a finite-dimensional real vector space. A function M: X k → X is called a strict mean value, if M(x1,…, xk) lies in the convex hull of x1,…, xk), but does not coincide with one of its vertices. A sequence (xn)n∈ ? in X is called M-recursive if xn+k = M(xn, xn+1,…, xn+k?1) for all n. We prove that for a continuous strict mean value M every M-recursive sequence is convergent. We give a necessary and sufficient condition for a convergent sequence in X to be M-recursive for some continuous strict mean value M, and we characterize its limit by a functional equation. 39 B 72, 39 B 52, 40 A 05. |
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