# Bound on approximation error of Bernstein polynomial

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 $$|B_n(x)-f(x)|$$?
• You should use probability, and Tchebycheff inequality and use$$|x-\frac{k}{n}|^\alpha \leq \frac{1}{n^{\alpha/2}}(1+\sqrt{n}|x-\frac{k}{n}|)$$
– EDX
Commented Oct 31, 2020 at 21:34

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 $$C^2$$

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

• Thank you for the references. Commented Nov 4, 2020 at 0:26
• @GiuseppeNegro Glad you found them useful! I have a few more, similar to these. If you don't find what you want in the ones I linked just let me know Commented Nov 4, 2020 at 11:36
• Do you know of references for non-uniform (but "rapid") convergence of functions by polynomials with Bernstein coefficients, especially where those coefficients all lie in [0, 1], or where the function is 0 or 1 at the points 0 and/or 1? See also my related question. Commented Nov 27, 2020 at 2:09
• @PeterO. Guan's idea of iterating (applying a Bernstein approximation to the error of the previous approximation) is nice, and improves convergence, but the resulting polynomial gives values outside $[0,1]$. For uniform bounds, the rate $1/n$ cannot be improved (see thereom 3 in Guan's paper). Abel, 2020 gives explicit, non-uniform bounds that improve previous results Commented Nov 27, 2020 at 9:20
• Thank you for responding. My interest is not so much about convergence rates as it is about algorithms to compute monotone sequences of polynomials that converge from above and from below to a function (again, see my related question). Also, I say "non-uniform convergence" to mean especially that the sequence of polynomials can have, for instance, the first Bernstein coefficient all equal to 0, when f(0) is 0, under certain cases (an example of this is Thomas and Blanchet 2012, especially Figure 2). Commented Nov 27, 2020 at 10:43