TY - JOUR
T1 - Interpolatory estimates for convex piecewise polynomial approximation
AU - Kopotun, K. A.
AU - Leviatan, D.
AU - Shevchuk, I. A.
N1 - Publisher Copyright:
© 2019 Elsevier Inc.
PY - 2019/6/1
Y1 - 2019/6/1
N2 - In this paper, among other things, we show that, given r∈N, there is a constant c=c(r) such that if f∈C r [−1,1] is convex, then there is a number N=N(f,r), depending on f and r, such that for n≥N, there are convex piecewise polynomials S of order r+2 with knots at the nth Chebyshev partition, satisfying |f(x)−S(x)|≤c(r)(min{1−x 2 ,n −1 1−x 2 }) r ω 2 (f (r) ,n −1 1−x 2 ), for all x∈[−1,1]. Moreover, N cannot be made independent of f.
AB - In this paper, among other things, we show that, given r∈N, there is a constant c=c(r) such that if f∈C r [−1,1] is convex, then there is a number N=N(f,r), depending on f and r, such that for n≥N, there are convex piecewise polynomials S of order r+2 with knots at the nth Chebyshev partition, satisfying |f(x)−S(x)|≤c(r)(min{1−x 2 ,n −1 1−x 2 }) r ω 2 (f (r) ,n −1 1−x 2 ), for all x∈[−1,1]. Moreover, N cannot be made independent of f.
KW - Convex approximation by polynomials
KW - Degree of approximation
KW - Jackson-type interpolatory estimates
UR - http://www.scopus.com/inward/record.url?scp=85060968288&partnerID=8YFLogxK
U2 - 10.1016/j.jmaa.2019.01.055
DO - 10.1016/j.jmaa.2019.01.055
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AN - SCOPUS:85060968288
SN - 0022-247X
VL - 474
SP - 467
EP - 479
JO - Journal of Mathematical Analysis and Applications
JF - Journal of Mathematical Analysis and Applications
IS - 1
ER -