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 -