TY - JOUR

T1 - Pointwise estimates for 3-monotone approximation

AU - Bondarenko, Andriy

AU - Leviatan, Dany

AU - Prymak, Andriy

N1 - Funding Information:
The first author was supported in part by the Ministerio de Ciencia e Innovación , Spain, grant MTM 2011-2763 . Part of this work was done while the second author visited the University of Manitoba. The third author was supported in part by NSERC of Canada .

PY - 2012

Y1 - 2012

N2 - We prove that for a 3-monotone function F ∈ C [ - 1, 1] , one can achieve the pointwise estimates |F(x)-Ψ(x)|≤cω3(F,ρn(x)),x∈[-1,1], where ρn(x){double colon equal}1/n 2+√1-x 2/n and c is an absolute constant, both with Ψ, a 3-monotone quadratic spline on the nth Chebyshev partition, and with Ψ, a 3-monotone polynomial of degree ≤. n.The basis for the construction of these splines and polynomials is the construction of 3-monotone splines, providing appropriate order of pointwise approximation, half of which nodes are prescribed and the other half are free, but "controlled".

AB - We prove that for a 3-monotone function F ∈ C [ - 1, 1] , one can achieve the pointwise estimates |F(x)-Ψ(x)|≤cω3(F,ρn(x)),x∈[-1,1], where ρn(x){double colon equal}1/n 2+√1-x 2/n and c is an absolute constant, both with Ψ, a 3-monotone quadratic spline on the nth Chebyshev partition, and with Ψ, a 3-monotone polynomial of degree ≤. n.The basis for the construction of these splines and polynomials is the construction of 3-monotone splines, providing appropriate order of pointwise approximation, half of which nodes are prescribed and the other half are free, but "controlled".

KW - 3-monotone approximation by piecewise polynomials and splines

KW - 3-monotone polynomial approximation

KW - Degree of pointwise approximation

KW - Shape preserving approximation

UR - http://www.scopus.com/inward/record.url?scp=84863448952&partnerID=8YFLogxK

U2 - 10.1016/j.jat.2012.06.002

DO - 10.1016/j.jat.2012.06.002

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AN - SCOPUS:84863448952

SN - 0021-9045

VL - 164

SP - 1205

EP - 1232

JO - Journal of Approximation Theory

JF - Journal of Approximation Theory

IS - 9

ER -