Bound on approximation error of Bernstein polynomial

Question

Consider a continuous function $f: [0,1] \to [0,1]$. Let $B_n$ be its $n$-th order Bernstein polynomial, $$ B_n(x) = \sum_{k=0}^n f\left(\frac{k}{n}\right) \binom{n}{k}x^k (1-x)^{n-k}. $$ As is well known, $B_n(x) \rightarrow f(x)$ uniformly on $[0,1]$ as $n \rightarrow \infty$. I am interested in bounding the approximation error $B_n(x)-f(x)$.

This reference, section 4, contains one such bound: $$ |B_n(x)-f(x)| \leq \left( 1 + \frac{1}{4n^2} \right) \omega(n^{-1/2}) $$ where $\omega$ is the modulus of continuity of $f$, that is, $\omega(\delta) = \sup_{|x-x'|<\delta} |f(x)-f(x')|$.

My questions are

• Is there any reference or proof to that result?
• Are there any similar results that provide a bound on $|B_n(x)-f(x)|$?

Answer 1

I found a few references:

• "Iterated Bernstein polynomial approximations", by Zhong Guan: theorem 1 mentions that if $f$ is $C^r$ for $r=0$ or $1$,

  $$|B_n(x)-f(x)| \leq C_rn^{-r/2}\omega_r(n^{-1/2})$$

  where $\omega_r$ is the modulus of continuity of the $r$-th derivative; and $C_0=5/4$, $C_1=3/4$. If $f$ is $C^r$ with $r>1$ the rate $1/n$ cannot be improved.

• "On the Rate of Approximation of Functions by the Bernstein Polynomials", by Telyakovskii , contains the above results and some refinemets for the error at a specific $x$.

• "Rate of convergence of Bernstein polynomials for functions with derivatives of bounded variation", by Bojanic and Cheng, improves $C_0$ from the above $5/4$ to

  $$\frac{4306 + 837\sqrt 6}{5832}$$.

• "Approximation of Hölder Continuous Functions by Bernstein Polynomials", by Mathé, establishes that

  $$|B_n(x)-f(x)| \leq L \left( \frac{x(1-x)}{n} \right)^{\alpha/2}$$

  when $f$ is Hölder with exponent $\alpha$ for some $0<\alpha\leq 1$ and constant $L$.

• "The Weierstrass Approximation Theorem and Large Deviations", by Gzyl and Palacios, mentions that if $f$ is $C^2$

  $$|B_n(x)-f(x)| \leq \frac{\sup_x|f''(x)|}{8n}.$$