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

1 Answer 1


I found a few references:

  • 1
    $\begingroup$ Thank you for the references. $\endgroup$ Commented Nov 4, 2020 at 0:26
  • $\begingroup$ @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 $\endgroup$
    – Luis Mendo
    Commented Nov 4, 2020 at 11:36
  • $\begingroup$ 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. $\endgroup$
    – Peter O.
    Commented Nov 27, 2020 at 2:09
  • $\begingroup$ @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 $\endgroup$
    – Luis Mendo
    Commented Nov 27, 2020 at 9:20
  • $\begingroup$ 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). $\endgroup$
    – Peter O.
    Commented Nov 27, 2020 at 10:43

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