Freeknot splines approximation of Sobolev-type classes of <Emphasis Type="Italic">s</Emphasis>-monotone functions |
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Authors: | V N Konovalov D Leviatan |
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Institution: | (1) Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv, 01601, Ukraine;(2) School of Mathematical Sciences, Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, 69978, Israel |
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Abstract: | Let I be a finite interval, s ∈ ℕ0, and r,ν,n ∈ ℕ. Given a set M, of functions defined on I, denote by
M the subset of all functions y ∈ M such that the s-difference is nonnegative on I, ∀τ > 0. Further, denote by the Sobolev class of functions x on I with the seminorm . Also denote by Σ
ν,n
, the manifold of all piecewise polynomials of order ν and with n – 1 knots in I. If ν ≥ max {r,s}, 1 ≤ p,q ≤ ∞, and (r,p,q) ≠ (1,1,∞), then we give exact orders of the best unconstrained approximation and of the best s-monotonicity preserving approximation .
Part of this work was done while the first author visited Tel Aviv University in May 2003 and in March 2004. |
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Keywords: | Shape preserving Free-knot spline Order of approximation |
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