The Bernstein Polynomials for Discontinuous Functions

On the Rate of Convergence of Two Bernstein–Bézier Type Operators for Bounded Variation Functions

“…Herzog and Hill [13] proved that if f is bounded on [0, 1] and x is a point of discontinuity of the first kind, then…”

Variation convergence for Bernstein polynomials

“…as in the paper of Herzog and Hill [2]. For functions/ and g, we have (1.3) | VBnf -VBng | g V(27/ -Bng) = WBn(J -g).…”

“…Preliminary notions. Most of the facts which will be needed can be found in [2] or [3]. In particular, we need the following:…”

On the variation of the Bernstein polynomials of a function of unbounded variation

“…It is shown in [2 p. 28] that B n f, for unbounded /, can behave unusually in terms of point wise convergence to /. Here we construct a function, unbounded on the rationals in every subinterval of [0,1], and which has the property that B n f converges in variation (and uniformly) to zero. 3* Construction* We define a sequence of skeletons /; such that each skeleton tends to + oo on a set of rationals tending to a limit rational r { .…”

“…3* Construction* We define a sequence of skeletons /; such that each skeleton tends to + oo on a set of rationals tending to a limit rational r { . The r, will be dense in [0,1]. It is shown that the skeleton /= Σ£=i/» has the following properties:…”