Approximation by Bernstein polynomials at the discontinuity points of derivatives (Q605981)

Let \(f \) be a given bounded function defined over \([0,1]\) and \(B_n f\) be its Bernstein polynomial of degree \(n\). The problem of obtaining an asymptotic formula for \((f - B_n f)(x) \) at \(x \in (0,1)\) has been studied here. Assuming that at \(x \in (0,1)\), \(f^{2r-1}\) exists for \(r=1,2,\dots\), and \(f^{2r}\) has a discontinuity of the first kind an asymptotic formula for \((f-B_n f)\) have been obtained in [\textit{S. A. Telyakovokii}, Math. Notes 85, No.~4, 590--596 (2009); translation from Mat. Zametki 85, No. 4, 622--629 (2009; Zbl 1183.41006)]. In the present paper similar estimates have been obtained for the case when odd order derivatives of \(f\) have a discontinuity of the first kind.