Freeknot splines approximation of Sobolev-type classes of s-monotone functions

Authors: Konovalov, V.¹; Leviatan, D.²

Source: Advances in Computational Mathematics, Volume 27, Number 2, August 2007 , pp. 211-236(26)

Publisher: Springer

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Abstract:

Let I be a finite interval, s ∈ ₀, and r,ν,n ∈ . Given a set M, of functions defined on I, denote by <EquationSource Format="TEX"><![CDATA[$Delta ^{s}_{ + } $]]></EquationSource> M the subset of all functions y ∈ M such that the s-difference <EquationSource Format="TEX"><![CDATA[$Delta ^{s}_{tau } y(cdot)$]]></EquationSource> is nonnegative on I, ∀τ > 0. Further, denote by <EquationSource Format="TEX"><![CDATA[$W^{r}_{p} $]]></EquationSource> the Sobolev class of functions x on I with the seminorm <EquationSource Format="TEX"><![CDATA[$|x^{(r)}|_{L_p}le 1$]]></EquationSource> . Also denote by Σ _ν,n, the manifold of all piecewise polynomials of order ν and with n - 1 knots in I. If ν ≥ max {r,s}, 1 ≤ p,q ≤ ∞, and (r,p,q) ≠ (1,1,∞), then we give exact orders of the best unconstrained approximation <EquationSource Format="TEX"><![CDATA[$Ebigl(Delta^s_+W^r_p,Sigma_{u,n}bigr)_{L_q}$]]></EquationSource> and of the best s-monotonicity preserving approximation <EquationSource Format="TEX"><![CDATA[$Ebigl(Delta^s_+W^r_p,Delta^s_+Sigma_{u,n}bigr)_{L_q}$]]></EquationSource> .

Keywords: Shape preserving; Free-knot spline; Order of approximation; 41A15; 41A25; 41A29

Document Type: Research article

DOI: http://dx.doi.org/10.1007/s10444-007-9032-9

Affiliations: 1: Email: vikono@imath.kiev.ua 2: Email: leviatan@math.tau.ac.il

Publication date: 2007-08-01