Widths and shape-preserving widths of Sobolev-type classes of s-monotone functions

نویسندگان

  • Jacek Gilewicz
  • Viktor N. Konovalov
  • Dany Leviatan
چکیده

Abstract. Let I be a finite interval, r, n ∈ N, s ∈ N0 and 1 ≤ p ≤ ∞. Given a set M , of functions defined on I, denote by ∆+M the subset of all functions y ∈ M such that the s-difference ∆τ y(·) is nonnegative on I, ∀τ > 0. Further, denote by W r p the Sobolev class of functions x on I with the seminorm ‖x‖Lp ≤ 1. We obtain the exact orders of the Kolmogorov and the linear widths, and of the shape-preserving widths of the classes ∆+W r p in Lq for s > r + 1 and (r, p, q) 6= (1, 1,∞). We show that while the widths of the classes depend in an essential way on the parameter s, which characterizes the shape of functions, the shape-preserving widths of these classes remain asymptotically 3 n−2.

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عنوان ژورنال:
  • Journal of Approximation Theory

دوره 140  شماره 

صفحات  -

تاریخ انتشار 2006