Distribution of fractional parts and approximation of functions with singularities by Bernstein polynomials (Q1095328)

The author establishes an asymptotic property of the sequence of fractional parts \(\{\) \(n\alpha\}\), (n\(\geq 1)\) for both rational and irrational values of \(\alpha\) and relates this with the asymptotic behaviour of the uniform approximation by Bernstein polynomials \((B_ nf)(x)=\sum^{n}_{k=0}\left( \begin{matrix} n\\ k\end{matrix} \right)x^ k(1- x)^{n-k}f(k/n)\) of functions f(x) having a singularity on the interval \(0\leq x\leq 1\). The main theorem of the paper is the following. Theorem: If \(f(x)=| x-\alpha |\), \(0<\alpha <1\), then \[ (B_ nf)(\alpha)=\sqrt{(2/\pi \alpha (1-\alpha))}\sum^{\infty}_{k=0}(C_ k(\alpha)/k^{3/2})+O(1/n) \] uniformly for \(n\geq 1\) and \(\alpha\in [\epsilon,1-\epsilon]\), where \(\epsilon\) is arbitrary.