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 -