T. Popoviciu, Sur les l’approximations des fonctions continues d’une variable réelle par des polynomes, Ann. Sci. Univ. Iassy, 28 (1942) p. 208 (in French).
give, for the functionf(x)f(x)continues in the closed interval[0,1][0,1], an approximation of the order ofomega((1)/(sqrtn))\omega\left(\frac{1}{\sqrt{n}}\right)ifomega (delta)omega(delta)is the oscillation modulus off(x)f(x)Here is a simpler demonstration of this result.
If we notice thatsum_(i=0)^(n)((n)/(i))x^(i)(1-x)^(ni)=1\sum_{i=0}^{n}\binom{n}{i} x^{i}(1-x)^{ni}=1the known properties ofomega (delta)omega(delta)give us
|f(x)-P_(n)(x;f)| <= ((1)/(delta)sum_(i=0)^(n)|x-(i)/(n)|((n)/(i))x^(i)(1-x)^(ni)+1)omega{delta},quad x in[0,1].\left|f(x)-P_{n}(x ; f)\right| \leq\left(\frac{1}{\delta} \sum_{i=0}^{n}\left|x-\frac{i}{n}\right|\binom{n}{i} x^{i}(1-x)^{ni}+1\right) \omega\{\delta\}, \quad x \in[0,1] .
But we have, by applying a well-known inequality,
{:[sum_(i=0)^(n)|x-(i)/(n)|((n)/(i))x^(i)(1-x)^(ni) <= sqrt(sum_(i=0)^(n)(x-(i)/(n))^(2)((n)/(i))^(x^(i)(1-x)^(ni)))=],[=sqrt((x(1-x))/(n)) <= (1)/(2sqrtn)","quad x in[0","1]]:}\begin{gathered} \sum_{i=0}^{n}\left|x-\frac{i}{n}\right|\binom{n}{i} x^{i}(1-x)^{ni} \leq \sqrt{\sum_{i=0}^{n}\left(x-\frac{i}{n}\right)^{2}\binom{n}{i}^{x^{i}(1-x)^{ni}}}= \\ =\sqrt{\frac{x(1-x)}{n}} \leq \frac{1}{2 \sqrt{n}}, \quad x \in[0,1] \end{gathered}
By takingdelta = (1)/(sqrtn)\delta=\frac{1}{\sqrt{n}}, We have
