关于可微函数类的Колмгоров宽度

ESTIMATION OF THE n-WIDTH OF SOME CLASSES OF DIFFERENTIABLE FUNCTIONS

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Let W^()L_p[-π,π]be the class of functions f,for which f^(r-1)(r=1,2,…) are absolutely continuous on the interval[-π,π]and ‖f^(r)‖p≤1(1≤p≤∞), and d_n[W^rL_p[-π,π],L_1]=inf sup inf‖f-g‖L_1,Where M_n()L_1 () [-π,π]is an arbitrary n-dimensional subspace of L_1[-π,π].The following theorem is established. Theorem 1. () Theorem 2.For 1≤p≤∞,r=1,2,3,……we have () where d_n~′[W^rL_∞,L_p]is the n-th linear width of the class W^rL_∞ with respect to L_p norm.