Let $f$ be a Holder continuous function in $[a,b]$,
i.e. $|f(x)-f(y)| \leq C_1 |x-y|^{\alpha}$ for some $\alpha \in (0,1)$.
and let $f_n$ be its Bernstein polynomial approximation.
Is it true that $f_n$'s are $\beta-$Holder continuous and $|f_n(x)-f_n(y)| \leq C_2 |x-y|^{\beta}$ in [a,b] with $C_2$ and $\beta \in (0,1)$ are independent of $n.$
Best Answer
Yes it is true. The proof (rather long), relying on a probabilistic argument, can be found in:
Mathe, Peter. “Approximation of Holder Continuous Functions by Bernstein Polynomials.” The American Mathematical Monthly, vol. 106, no. 6, 1999, pp. 568–574. JSTOR, www.jstor.org/stable/2589469.
It is proposition 2 (last 2 pages of the article).