This paper examines the problem of finding the linear algorithm (operator) of finite rank n (i.e. with a n-dimensional range) which gives the minimal error of approximation of identity operator on some set over all finite rank n linear operators preserving the cone of k-monotonicity functions. We introduce the notion of linear relative (shape-preserving) n-width and find asymptotic estimates of linear relative n-widths for linear operators preserving k-monotonicity in the space C~k[0,1]. The estimates show that if linear operator with finite rank n preserves k-monotonicity, the degree of simultaneous approximation of derivative of order 0 ≤ i ≤ k of continuous functions by derivatives of this operator cannot be better than n~(-2) even on the set of algebraic polynomials of degree k + 2 (as well as on bounded subsets of Sobolev space W_∞~((k+2)) [0,1]).
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