Shape-Preserving Widths of Weighted Sobolev-Type Classes of Positive, Monotone, and Convex Functions on a Finite Interval |
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Authors: | Konovalov Leviatan |
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Institution: | International Mathematical Center National Academy of Sciences of Ukraine Kyiv 01601 Ukraine, UA School of Mathematical Sciences Sackler Faculty of Exact Sciences Tel Aviv University Tel Aviv 69978 Israel and IMI Department of Mathematics University of South Carolina Columbia, SC 29208 USA, US
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Abstract: | Abstract. Let I be a finite interval, r∈ N and ρ(t)= dist {t, I} , t∈ I . Denote by Δ s + L q the subset of all functions y∈ L q such that the s -difference Δ s τ y(t) is nonnegative on I , $$\forall$$ τ>0 . Further, denote by $$\Delta^s_+W^r_{p,\alpha}$$ , 0≤α<∞ , the classes of functions x on I with the seminorm ||x (r) ρ α ||_ L p ≤ 1 , such that Δ s τ x≥ 0 , τ>0 . For s=0,1,2 , we obtain two-sided estimates of the shape-preserving widths $$d_n (\Delta _ + ^s W_{p,\alpha ,}^r \Delta _ + ^s L_q )L_q : = \mathop {\inf }\limits_{M^n \in \mathcal{M}^n } \mathop {\sup }\limits_{x \in \Delta _ + ^s W_{p,\alpha }^r } \mathop {\inf }\limits_{y \in M^n \cap \Delta ^s + L_q } \left\| {x - y} \right\|L_q$$ where M n is the set of all linear manifolds M n in L q such that dim M n ≤ n , and satisfying $$M^n\cap\Delta^s_+L_q\neq\emptyset$$ . |
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