Abstract
Let s ∈ ℕ0 and let Δ s+ be the set of functions x defined on a finite interval I and such that, for all collections of s + 1 pairwise different points t 0,..., t s ∈ I , the corresponding divided differences [x; t 0 ,..., t s ] of order s are nonnegative. Let Δ s+ B p ≔ Δ s+ ⋂ B p, 1 ≤ p ≤ ∞ where B p is a unit ball in the space L p , and let Δ s+ L q ≔ Δ s+ ⋂ L q, 1 ≤ q ≤ ∞. For every s ≥ 3 and 1 ≤ q ≤ p ≤ ∞, we determine the exact orders of the shape-preserving Kolmogorov widths <Equation ID=”IE4”><EquationSource Format=”MATHTYPE”><![CDATA[%MathType!MTEF!2!1!+%feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn%hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8qqaq% Fr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qq%Q8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatC%vAUfKttLearyqr1ngBPrgaiuGacqWFKbazcqWFUbGBdaqadaqaaiab% gs5aenaaDaaaleaacqGHRaWkaeaacqWFZbWCaaGccqWFcbGqdaWgaa%WcbaGae8hCaahabeaakiaacYcacqGHuoardaqhaaWcbaGaey4kaSca%baGae83CamhaaOGae8htaW0aaSbaaSqaaiab-bhaWbqabaaakiaawI%cacaGLPaaadaqhaaWcbaGae8htaW0aaSbaaWqaaiab-fhaXbqabaaa% leaaiuaacqGFRbWAcqGFVbWBcqGFSbaBaaGccaGG6aGaeyypa0ZaaC%beaeaaciGGPbGaaiOBaiaacAgaaSqaaiab-1eannaaCaaameqabaGa%e8NBa4gaaSGaeyicI48efv3ySLgznfgDOfdarCqr1ngBPrginfgDOb% YtUvgaiyaacqqFZestdaahaaadbeqaaiab-5gaUbaaaSqabaGcdaWf%qaqaaiGacohacaGG1bGaaiiCaaWcbaGae8hEaGNaeyiLdq0aa0baaW% qaaiabgUcaRaqaaiab-nhaZbaaliab-jeacnaaBaaameaacqWFWbaC% aeqaaaWcbeaakiab-Lha5jabgIGiolab-1eannaaCaaaleqabaWaaC%beaeaaciGGPbGaaiOBaiaacAgaaWqaaiab-5gaUjabgMIihlabgs5a% enaaDaaabaGaey4kaScabaGae83Camhaaiab-XeamnaaBaaabaGae8% CaahabeaaaeqaaaaakmaafmaabaGae8hEaGNaeyOeI0Iae8xEaKha% caGLjWUaayPcSdGae8htaW0aaSbaaSqaaiab-fhaXbqabaGccaGGSa% aaaa!923E! ]]></EquationSource><EquationSource Format=”TEX”><![CDATA[$$dn\left( {\Delta _ + ^s B_p ,\Delta _ + ^s L_p } \right)_{L_q }^{kol} : = \mathop {\inf }\limits_{M^n \in \mathcal{M}^n } \mathop {\sup }\limits_{x\Delta _ + ^s B_p } y \in M^{\mathop {\inf }\limits_{n \cap \Delta _ + ^s L_p } } \left\”>{x - y} \right\ L_q , $$]]></EquationSource></Equation>, where M n is the collection of all affine linear manifolds M n in L q such that dim M n ≤ n and M n ⋂ Δ s+ L q ≔ ∅.
Similar content being viewed by others
REFERENCES
A. N. Kolmogoroff (1936) ArticleTitleUber die beste Annäherung von Funktionen einer gegebenen Funktionenklasse Math. Ann. 37 107–110
V. N. Konovalov (1984) ArticleTitleEstimates of Kolmogorov widths for classes of differentiable periodic functions Mat. Zametki 35 IssueID3 369–380
P. S. Bullen (1971) ArticleTitleA criterion for n-convexity Pacif. J. Math. 36 81–98
A. W. Roberts D. E. Varberg (1973) Convex Functions Academic Press New York
J. E. Pečarić F. Proschan Y. L. Tong (1992) Math. Sci. Eng. Convex Functions, Partial Orderings, and Statistical Applications Academic Press Boston
V. N. Konovalov D. Leviatan (2003) ArticleTitleShape-preserving widths of weighted Sobolev-type classes of positive, monotone and convex functions on a finite interval Constr. Approxim. 19 23–58
V. N. Konovalov D. Leviatan (2003) ArticleTitleShape preserving widths of Sobolev-type classes of s-monotone functions on a finite interval Isr. J. Math. 133 239–268
V. N. Konovalov D. Leviatan (2001) ArticleTitleEstimates on the approximation of 3-monotone functions by 3-monotone quadratic splines East J. Approxim. 7 333–349
V. N. Konovalov (2004) ArticleTitleShape preserving widths of Kolmogorov type of the classes of positive, monotone, and convex integrable functions East J. Approxim. 10 IssueID1-2 93–117
E. D. Gluskin (1986) ArticleTitleOctahedron is poorly approximated by random subspaces Funkts. Anal. Prilozhen. 20 IssueID1 14–20
A. Y. Garnaev E. D. Gluskin (1984) ArticleTitleOn the widths of Euclidean ball Dokl. Akad. Nauk SSSR 277 IssueID5 1048–1052
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 7, pp. 901–926, July, 2004.
Rights and permissions
About this article
Cite this article
Konovalov, V.N. Shape-preserving kolmogorov widths of classes of s-monotone integrable functions. Ukr Math J 56, 1074–1101 (2004). https://doi.org/10.1007/PL00022180
Received:
Issue Date:
DOI: https://doi.org/10.1007/PL00022180