Abstract
Let s ∈ ℕ and let Δ s+ be the set of functions x: I ↦ ℝ on a finite interval I such that the divided differences [x; t 0, ..., t s ] of order s of these functions are nonnegative for all collections of s + 1 different points t 0, ..., t s ∈ I. For the classes Δ s+ B p : = Δ s+ ∩ B p , where B p is the unit ball in L p , we determine the orders of Kolmogorov and linear widths in the spaces Lq for 1 ≤ q > p ≤ ∞.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
P. S. Bullen, “A criterion for n-convexity,” Pacif. J. Math., 36, 81–98 (1971).
A. W. Roberts and D. E. Varberg, Convex Functions, Academic Press, New York (1973).
J. E. Pečarić, F. Proschan, and Y. L. Tong, Convex Functions, Partial Orderings, and Statistical Applications, Academic Press, Boston (1992).
V. N. Konovalov, “Shape-preserving Kolmogorov widths of classes of s-monotone integrable functions,” Ukr. Mat. Zh., 55, No. 7, 901–926 (2004).
V. N. Konovalov, “Shape preserving widths of Kolmogorov type of the classes of positive, monotone, and convex integrable functions,” E. J. Approxim., 10, No. 1–2, 93–117 (2004).
V. N. Konovalov and D. Leviatan, “Kolmogorov and linear widths of weighted Sobolev-type classes on a finite interval. II,” J. Approxim. Theory, 113, 266–297 (2001).
V. N. Konovalov and D. Leviatan, “Shape-preserving widths of weighted Sobolev-type classes of positive, monotone and convex functions on a finite interval,” Constr. Approxim., 19, 23–58 (2003).
V. N. Konovalov and D. Leviatan, “Shape-preserving widths of Sobolev-type classes of s-monotone functions on a finite interval,” Isr. J. Math., 133, 239–268 (2003).
R. A. de Vore and G. G. Lorentz, Constructive Approximation, Springer, Berlin (1993).
E. D. Gluskin, “Norms of random matrices and widths of finite-dimensional sets,” Mat. Sb., 120, No. 1, 180–189 (1986).
Author information
Authors and Affiliations
Additional information
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 12, pp. 1633–1652, December, 2005.
Rights and permissions
About this article
Cite this article
Konovalov, V.N. Kolmogorov and linear widths of classes of s-monotone integrable functions. Ukr Math J 57, 1911–1936 (2005). https://doi.org/10.1007/s11253-006-0039-z
Received:
Issue Date:
DOI: https://doi.org/10.1007/s11253-006-0039-z