By B. Sendov (auth.), Gerald Beer (eds.)
ISBN-10: 9400906730
ISBN-13: 9789400906730
ISBN-10: 9401067872
ISBN-13: 9789401067874
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Sample text
IT f, g E An then by definition r(o, a; f, g) =rea; f, g) =rea; F(t), F(g». Of course, the H-distance in An does not meet all the axioms of the metric. The condition rea; f, g) =0 ~ f = g is violated, since two different functions may have the same completed graphs. We shall consider some properties of H-distance in An and in Fn. First, we observe that from the inclusion Al c:: A2 for intervals, it does not follow that r(A}, a; f, g) S r(A2, a; f, g). 2) that r([-1, 1], a; cp, 'If) = 1 whereas r([-2, 2], a; cp, 'If) = 1/(1 + a) < 1.
Proof. 12), since r(a; f, g) is at most the Hausdorff distance between the graphs of f and g, and for y e f(x) and 11 e g(x), we have inf max {a-1Ix - ~I, Iy -11'I} ~ Iy -111 ~ R(f,g). 44), we may assume that f and g are in Fa, as Hausdorff distance between segment functions is defined in tenns of completed graphs, and unifonn distance is not decreased by replacing segment functions by their completed graphs. It is directly seen that for every x e n and y e f(x) there exists . 12), for every x e a-1lx - ~I ~ r(a; f, g), Iy - TI'I ~ =max n { a-1lx - ~xl, Iy - TI'I}.
Markov [2]. A definition of derivative with respect to Hausdorff distance was given by Bl. Sendov, B. Penkov, V. Popov and S. Markov [1]. A detailed characterization of the segment derivative of a Lipschitz function is given in Bl. Sendov, S. Tashev, and P. Petrushev [1]. A sufficient condtion for the existence of a solution of a differential equation, that involves a segment derivative is considered in S. Tashev [2]. Segment derivation of a function is used in S. Tashev [4]. The notion of the complete graph of a function is used in many situations, even when Hausdorff distance is not considered.
Hausdorff Approximations by B. Sendov (auth.), Gerald Beer (eds.)
by Daniel
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