A note on a result of Bernstein
The author proves that for any \(a\geq 2\), the function \((1-x)^{-1}\) can be approximated on [-1/a,1/a] by polynomials of degree n, having nonnegative, nonincreasing coefficients with an error \(<a(a^{n+1}- 1)^{-1}\); but never better than \((a^{n+1}-1)^{-1}\). It is also proved that the smallest maximal error in a uniform approximation of 1-x on [0,1] by ratios of polynomials of degree \(\leq n\), \(n\geq 0\), having nonnegative, nonincreasing coefficients is \((n+2)^{-1}\).
scientific article; zbMATH DE number 4017531