Weak interpolation and approximation of non-linear operators on the space C([0,1])
Date
1998
Authors
Torokhti, A.
Howlett, P.G.
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Numerical Functional Analysis and Optimization, 1998; 19(9-10):1025-1043
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Abstract
In this paper we define and discuss a new non-Lagrangean procedure for the weak interpolation of a non-linear mapping on the space of continuous functions. We suppose that the mapping F : C([0,1]) → C([0,1]) is defined by an empirical data set D<inf>p</inf> = {(x<inf>r</inf>, y<inf>r</inf>)| y<inf>r</inf> = F[x<inf>r</inf>] for each r = 1, 2,... ,p} C C([0,1]) × C([0, 1]) where the functions x<inf>r</inf> and y<inf>r</inf> are known only by evaluation vectors (x<inf>r</inf>(s<inf>t</inf>)) e R<sup>m</sup> and (y<inf>r</inf>(t<inf>k</inf>)) ∈ R<sup>n</sup>. In this situation the Lagrangean interpolation proposed by Prenter cannot be applied. We construct a mapping S : C[0,1] → C[0, 1] such that S[u] = S[x] When u(s<inf>t</inf>) = i(s<inf>t</inf>) for each i = 1,2, ...,m and such that S[x<inf>r</inf>](t<inf>k</inf>) = y<inf>r</inf>(t<inf>k</inf>) for each r = 1, 2,... , p and each K = 1, 2,. .. , n. We show that in some special circumstances the weak interpolation becomes a strong interpolation and we also show that the interpolation operator is continuous in the strong topology at each point of the empirical data set.
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