Comparison of Bernstein polynomials with the metrical means of Kantorovitch

AbstractIn 1934 Kantorovitch modified the Bernstein polynomials Bn by means of metrical means to yield a nonlinear polynomial process Bn∗ which approximates measurable functions almost everywhere. The present paper is concerned with the pointwise comparison of Bn and Bn∗ on C[0, 1] (the space of continuous functions on [0, 1]). We establish direct estimates of the form ¦(Bnf − f)(x)¦ ⩽ ¦(Bn∗f − f)(x)¦ + ω(3n−13, f) with the first modulus of continuity ω. On the other hand, it is the main purpose of this paper to show that this inequality can not be strengthened to ¦(Bnf − f)(x)¦ ⩽ Cx ¦(Bnf − f)(x)¦ so that Bn∗ is not a pointwise extension of Bn on C[0, 1]. To this end, a previous condensation principle is applied concerning nonlinear functionals which assures the nonvalidity of this last inequality for x on a dense, in fact, residual, set of [0, 1]