Shape-preserving kolmogorov widths of classes of s-monotone integrable functions

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 t0,..., ts ∈ I, the corresponding divided differences [x; t0,...,ts] of order s are nonnegative. Let Δ+sBp ≔ Δ+s ⋂ Bp, 1 ≤ p ≤ ∞ where Bp is a unit ball in the space Lp, and let Δ+sLq ≔ Δ+s ⋂ Lq, 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 Mn is the collection of all affine linear manifolds Mn in Lq such that dim Mn ≤ n and Mn ⋂ Δ+sLq ≔ ∅.