TY - JOUR
T1 - Positive Results and Counterexamples in Comonotone Approximation
AU - Leviatan, D.
AU - Radchenko, D. V.
AU - Shevchuk, I. A.
PY - 2012/10
Y1 - 2012/10
N2 - We estimate the degree of comonotone polynomial approximation of continuous functions f, on [-1,1], that change monotonicity s≥1 times in the interval, when the degree of unconstrained polynomial approximation E n(f)≤n -α, n≥1. We ask whether the degree of comonotone approximation is necessarily ≤c(α,s)n -α, n≥1, and if not, what can be said. It turns out that for each s≥1, there is an exceptional set A s of α's for which the above estimate cannot be achieved.
AB - We estimate the degree of comonotone polynomial approximation of continuous functions f, on [-1,1], that change monotonicity s≥1 times in the interval, when the degree of unconstrained polynomial approximation E n(f)≤n -α, n≥1. We ask whether the degree of comonotone approximation is necessarily ≤c(α,s)n -α, n≥1, and if not, what can be said. It turns out that for each s≥1, there is an exceptional set A s of α's for which the above estimate cannot be achieved.
KW - Comonotone polynomial approximation
KW - Constants in constrained approximation
KW - Degree of approximation
KW - Degree of comonotone approximation
UR - http://www.scopus.com/inward/record.url?scp=84865616072&partnerID=8YFLogxK
U2 - 10.1007/s00365-012-9159-x
DO - 10.1007/s00365-012-9159-x
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AN - SCOPUS:84865616072
SN - 0176-4276
VL - 36
SP - 243
EP - 266
JO - Constructive Approximation
JF - Constructive Approximation
IS - 2
ER -