On some constants in approximation by Bernstein operators. (English) Zbl 1199.41136
Summary: We estimate the constants
\[
\sup_{x\in (0,1)} \sup_{f\in C[0,1]\setminus {\Pi}_1} \frac {| B_n (f,x)-f(x)|}{{\omega}_2 (f,\sqrt{\frac {x(1-x)}n})}
\]
and
\[
\inf_{x\in (0,1)} \sup_{f\in C[0,1]\setminus {\Pi}_1} \frac {| B_n (f,x)-f(x)|}{{\omega}_2 (f,\sqrt{\frac {x(1-x)}n})},
\]
where \(B_n\) is the Bernstein operator of degree \(n\) and \({\omega}_2\) is the second order modulus of continuity.
MSC:
| 41A36 | Approximation by positive operators |
| 41A10 | Approximation by polynomials |
| 41A25 | Rate of convergence, degree of approximation |
| 41A35 | Approximation by operators (in particular, by integral operators) |
