Abstract
n-widths in approximation theory characterize how well one can approximate a subset by some “good” subsets of a normed linear space. Especially, n-widths of sets of mathbb{R}^{N} have been studied deeply. Now the following problem is posed: we know that mathbb{R} ^{N} can be embedded in the fuzzy number space E^{N}. Is it then possible to define n-widths of set A in E^{N} and obtain asymptotic estimates for these n-widths?In this paper, we shall introduce four n-widths of A in E^{N} and determine these n-widths of Zadeh’s extension of diagonal matrices.
Highlights
Let A and B be two subsets of a normed linear space X
One may ask: how well A can be approximated by B? In the theory of n-widths of A in X, B will be a simple subspace of X
Where Xn+1 is any (n + 1)-dimensional subspace of X, and S(Xn+1) is the unit ball of Xn+1 . (3) The Gelfand n-width of A in X is defined as dn(A; X) = inf sup x, Ln x∈A∩Ln where the infimum is taken over all subspaces Ln of X of codimension n
Summary
Let A and B be two subsets of a normed linear space X. 2.2 n-Widths of diagonal matrix Definition 1 ([10]) Let (X, · ) be a normed linear space, and A ⊆ X. (3) The Gelfand n-width of A in X is defined as dn(A; X) = inf sup x , Ln x∈A∩Ln where the infimum is taken over all subspaces Ln of X of codimension n. (4) The linear n-width is given by δn(A; X) = inf sup x – Pn(x) , Pn(A) x∈A
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