TY - JOUR
T1 - Rate of convergence of Bernstein polynomials for functions with derivatives of bounded variation
AU - Bojanic, Ranko
AU - Cheng, Fuhua
PY - 1989/7
Y1 - 1989/7
N2 - For any x ε{lunate} (0, 1) we first prove that if f{hook}x(t) ≡ |t - x| on [0, 1] then the Bernstein polynomials of f{hook}x satisfy the asymptotic relation ∑k = 0n | k n - x|(kn) xk(1 - x) n - k = (2x (1 - x) π) 1 2 1 √n + O( 1 n). This asymptotic relation is then used to study the rate of convergence of Bernstein polynomials for functions with derivatives of bounded variation. An estimate of the rate of convergence is given. This estimate is asymptotically the best possible at points where f{hook}′ is continuous.
AB - For any x ε{lunate} (0, 1) we first prove that if f{hook}x(t) ≡ |t - x| on [0, 1] then the Bernstein polynomials of f{hook}x satisfy the asymptotic relation ∑k = 0n | k n - x|(kn) xk(1 - x) n - k = (2x (1 - x) π) 1 2 1 √n + O( 1 n). This asymptotic relation is then used to study the rate of convergence of Bernstein polynomials for functions with derivatives of bounded variation. An estimate of the rate of convergence is given. This estimate is asymptotically the best possible at points where f{hook}′ is continuous.
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U2 - 10.1016/0022-247X(89)90211-4
DO - 10.1016/0022-247X(89)90211-4
M3 - Article
AN - SCOPUS:38249021452
SN - 0022-247X
VL - 141
SP - 136
EP - 151
JO - Journal of Mathematical Analysis and Applications
JF - Journal of Mathematical Analysis and Applications
IS - 1
ER -