Geodesic
On the extension and smoothing of vector-valued functions
Izvestiya. Mathematics , Tome 59 (1995) no. 4, pp. 847-879

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Smoothing of maps in Banach spaces is considered in this article. We construct an example of an infinitely differentiable vector-valued function on a subspace $L$ in $C[0,1]$ that does not have a uniformly continuous extension to a neighbourhood of $L$. The Kolmogorov widths obtained are correct in the order of growth of three parameters. 
@article{IM2_1995_59_4_a9,
     author = {I. G. Tsar'kov},
     title = {On the extension and smoothing of vector-valued functions},
     journal = {Izvestiya. Mathematics },
     pages = {847--879},
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     volume = {59},
     number = {4},
     year = {1995},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1995_59_4_a9/}
}
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I. G. Tsar'kov. On the extension and smoothing of vector-valued functions. Izvestiya. Mathematics , Tome 59 (1995) no. 4, pp. 847-879. http://geodesic.mathdoc.fr/item/IM2_1995_59_4_a9/